#72320004

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#359487
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#62636	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ C₂×#1757 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5628 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:

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

Polycyclic presentation:

2^10|a^2=e,d^2=j,e^2=ij,f^2=i,g^2=j,i^2=j,b^a=bf,c^a=cfgj,c^b=cf,d^a=dh,d^c=dj,e^b=ei,e^c=eij,f^a=fj,f^b=fij,f^c=fij,g^b=gj,g^c=gj,g^d=gj,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320043, #72320639, #72343980
maximal normal subgroup of	#72320043, #72320639, #72343980

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