#72317495

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#9497
commutator	C₂⁴×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#120014
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=f,c^2=fij,f^2=j,b^a=be,c^a=cfij,c^b=cg,d^b=dh,e^a=eh,e^c=ei,g^a=gi,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317506, #72317617, #72317779
maximal normal subgroup of	#72317506, #72317617, #72317779

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