#72341106

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#251884	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,13)(4,9,6,11)(5,10,20,26)(7,12)(8,16)(14,21,17,23)(15,22,30,27)(18,24)(19,25)(28,32)(29,31)(33,34)(35,44)(36,49,37,41)(38,42)(39,43)(40,48)(45,54,46,56)(47,55)(50,58,51,57)(52,60)(53,59)(61,64,62,63);(1,3)(2,8)(4,17,6,14)(5,15,20,30)(7,16)(9,23,11,21)(10,22,26,27)(12,13)(18,28)(19,29)(24,31)(25,32)(33,35)(34,40)(36,45,37,46)(38,52)(39,48)(41,56,49,54)(42,55)(43,44)(47,53)(50,62,51,61)(57,64,58,63)(59,60);(1,4)(2,9,12,11)(3,14)(5,19,20,18)(6,7)(8,21,13,23)(10,25)(15,28,30,29)(16,17)(22,31)(24,26)(27,32)(33,36)(34,41)(35,45,48,46)(37,39)(38,50,53,51)(40,54,44,56)(42,58,59,57)(43,49)(47,61)(52,62)(55,64)(60,63);(1,5)(2,10,12,26)(3,15,16,30)(4,18,6,19)(7,20)(8,22)(9,24)(11,25)(13,27)(14,28)(17,29)(21,31,23,32)(33,37,39,36)(34,41)(35,46,48,45)(38,50)(40,54)(42,57,59,58)(43,49)(44,56)(47,61)(51,53)(52,62)(55,63,60,64);(1,6,7,4)(2,11,12,9)(3,14)(5,19,20,18)(8,23)(10,24,26,25)(13,21)(15,29)(16,17)(22,32)(27,31)(28,30)(33,38)(34,42,43,59)(35,47,48,52)(36,50,37,51)(39,53)(40,55)(41,57)(44,60)(45,61)(46,62)(49,58)(54,63,56,64)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=gh,d^2=fj,e^2=fhij,b^a=bgi,c^a=cf,c^b=cfh,d^a=dfg,d^b=dfghj,d^c=dfghi,e^a=eghi,e^b=ef,e^c=efgij,e^d=efgi,f^a=fi,f^b=fij,f^c=fij,f^e=fi,g^c=gi,g^d=gi,g^e=gij,h^a=hij,h^b=hj,h^c=hi,h^d=hj,h^e=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341110, #72341113
maximal normal subgroup of	#72341110, #72341113

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);