#72319975

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#368835
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#7622935	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #10711 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2598 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)(3,10)(4,21)(5,9)(6,15)(7,14)(8,12)(11,25)(13,27)(16,32)(17,39)(18,38)(19,43)(20,45)(22,36)(23,41)(24,42)(26,40)(28,44)(29,35)(30,34)(31,37)(33,55)(46,59)(47,58)(48,61)(49,63)(50,57)(51,62)(52,53)(54,64)(56,60);(2,9)(3,16)(5,27)(6,7)(11,33)(12,34)(13,15)(14,35)(17,46)(18,47)(19,20)(21,41)(22,51)(23,55)(25,26)(28,56)(29,30)(31,48)(32,43)(37,45)(39,63)(40,60)(42,44)(49,54)(50,53)(57,59)(58,61)(62,64);(1,3)(2,10)(4,22)(5,16)(6,20)(7,31)(8,18)(9,32)(11,39)(12,38)(13,43)(14,37)(15,45)(17,26)(19,27)(21,52)(23,51)(24,53)(25,49)(28,50)(29,48)(30,47)(33,59)(34,58)(35,61)(36,42)(40,63)(41,57)(44,62)(46,56)(54,55)(60,64);(1,4)(2,11,12,40)(3,17,18,49)(5,28)(6,25)(7,26)(8,24)(9,33,34,60)(10,36)(13,44,35,41)(14,21,15,42)(16,46,47,54)(19,51,48,50)(20,22,31,53)(23,30)(27,56)(29,55)(32,57)(37,39)(38,52)(43,59)(45,63)(58,62)(61,64)

Polycyclic presentation:

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

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320043, #72320989, #72343971
maximal normal subgroup of	#72320043, #72320989, #72343971

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