#72341022

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

2256 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)(4,8,17,26)(5,18,21,9)(6,10)(7,14)(12,22,15,24)(13,23,19,28)(16,25)(20,27)(29,31)(30,32)(33,34)(35,45)(36,50,53,41)(37,54,52,42)(38,43)(39,44)(40,49)(46,61,64,55)(47,56,63,59)(48,57)(51,58)(60,62);(1,3)(2,7)(4,15,17,12)(5,19,21,13)(6,14)(8,24,26,22)(9,23,18,28)(10,11)(16,30)(20,29)(25,31)(27,32)(33,35)(34,40)(36,46,53,64)(37,47,52,63)(38,48)(39,49)(41,55,50,61)(42,59,54,56)(43,60)(44,45)(51,62)(57,58);(1,3)(2,7,10,11)(4,15)(5,13,21,19)(6,14)(8,24,26,22)(9,23)(12,17)(16,30,20,29)(18,28)(25,32)(27,31)(33,36,39,53)(34,41,44,50)(35,46)(37,51)(38,52)(40,55)(42,58)(43,54)(45,61)(47,62,63,48)(49,64)(56,60,59,57);(1,4,6,17)(2,8)(3,12)(5,20,21,16)(7,22,11,24)(9,27)(10,26)(13,29)(14,15)(18,25)(19,30)(23,31,28,32)(33,37,39,52)(34,42)(35,47,49,63)(36,51)(38,53)(40,56)(41,58,50,43)(44,54)(45,59)(46,62)(48,64)(55,60,61,57);(1,5)(2,9)(3,13,14,19)(4,16,17,20)(6,21)(7,23,11,28)(8,25,26,27)(10,18)(12,29)(15,30)(22,31)(24,32)(33,38,39,51)(34,43)(35,48)(36,52,53,37)(40,57,45,60)(41,42)(44,58)(46,63)(47,64)(49,62)(50,54)(55,56,61,59)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=hj,d^2=fghi,e^2=fj,b^a=bgi,c^a=cf,c^b=cgh,d^a=dfhj,d^b=dfi,d^c=dg,e^a=eghj,e^b=efhi,e^c=efgj,e^d=efg,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	#72341133
maximal normal subgroup of	#72341133

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