#72320863

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#377115
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#498872	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #10711 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2268 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efij,d^2=e,f^2=i,i^2=j,b^a=bf,c^a=cfh,c^b=cfi,d^a=dg,d^b=di,d^c=di,e^b=ej,e^c=ej,f^a=fi,f^b=fj,f^c=fij,f^d=fj,g^c=gj,h^d=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320921, #72320989, #72343936
maximal normal subgroup of	#72320921, #72320989, #72343936

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