#72225014

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#499962
commutator, Frattini, cosocle	C₂²×C₁₆	C₂⁴
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ #33433 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=gj,f^2=hij,h^2=ij,i^2=j,b^a=bfg,d^a=de,d^b=dgj,d^c=dgj,e^a=ej,e^c=ej,f^b=fh,f^d=fj,g^a=gj,h^b=hi,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225019, #72342984
maximal normal subgroup of	#72225019, #72342984

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