#72341005

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

2258 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)(3,11)(4,15,18,8)(5,9,20,25)(6,10)(7,14)(12,23,16,21)(13,26,19,22)(17,24)(27,31)(28,32)(29,30)(33,34)(35,45)(36,50,52,41)(37,42,54,60)(38,55)(39,44)(40,49)(43,51)(46,61,64,57)(47,62,53,58)(48,63)(56,59);(1,3)(2,7)(4,16,18,12)(5,19,20,13)(6,14)(8,23,15,21)(9,22,25,26)(10,11)(17,28)(24,31)(27,30)(29,32)(33,35)(34,40)(36,46,52,64)(37,53,54,47)(38,56)(39,49)(41,57,50,61)(42,58,60,62)(43,59)(44,45)(48,51)(55,63);(1,3)(2,7,10,11)(4,16)(5,13,20,19)(6,14)(8,23,15,21)(9,22)(12,18)(17,28,30,27)(24,32)(25,26)(29,31)(33,36)(34,41)(35,46,49,64)(37,38,54,51)(39,52)(40,57,45,61)(42,43,60,55)(44,50)(47,48)(53,56)(58,59)(62,63);(1,4,6,18)(2,8)(3,12)(5,17,20,30)(7,21,11,23)(9,24)(10,15)(13,28)(14,16)(19,27)(22,32,26,31)(25,29)(33,37,39,54)(34,42)(35,47,49,53)(36,51)(38,52)(40,58)(41,55,50,43)(44,60)(45,62)(46,56)(48,64)(57,63,61,59);(1,5,6,20)(2,9,10,25)(3,13)(4,17)(7,22)(8,24)(11,26)(12,27,16,28)(14,19)(15,29)(18,30)(21,31,23,32)(33,38,39,51)(34,43)(35,48)(36,37,52,54)(40,59,45,63)(41,60)(42,50)(44,55)(46,47)(49,56)(53,64)(57,62,61,58)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=h,d^2=fghi,e^2=fi,b^a=bgi,c^a=cf,c^b=cgh,d^a=dfhij,d^b=dfij,d^c=dgj,e^a=eghij,e^b=efhij,e^c=efg,e^d=efgi,f^a=fj,f^b=fi,f^c=fij,f^d=fij,g^c=gj,g^d=gi,g^e=gij,h^a=hij,h^b=hj,h^d=hj,h^e=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341033, #72341110
maximal normal subgroup of	#72341033, #72341110

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.8);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.6*F.9);;
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.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
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, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);