#72320191

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	#376962
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#13334	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #10711 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2257 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efgij,c^2=efgj,d^2=efgij,e^2=i,f^2=i,h^2=j,i^2=j,b^a=bf,c^a=cfgi,c^b=cfgj,d^a=dh,d^c=di,e^b=ei,e^c=eij,e^d=ej,f^a=fj,f^b=fi,f^c=fi,g^d=gj,h^c=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320238, #72320350, #72343980
maximal normal subgroup of	#72320238, #72320350, #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, 2, F.5*F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 3, F.5*F.6*F.7*F.10);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, 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.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
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);;
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);