#503275

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1065 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6)(7,20)(8,15)(9,31)(10,37)(11,18)(12,41)(13,19)(14,23)(16,33)(17,21)(22,53)(24,56)(25,47)(26,63)(27,45)(28,55)(29,49)(30,50)(32,54)(34,65)(35,42)(36,66)(38,51)(39,57)(40,43)(44,58)(46,62)(48,59)(52,61)(60,67)(64,68);(5,7,21,58,12,24,38,62,13,25,32,27,11,39,10,23)(6,14,37,57,18,45,54,47,19,46,51,56,41,44,17,20)(8,26,59,35,29,22,60,36,30,61,64,34,40,33,28,9)(15,31,55,16,43,65,68,52,50,66,67,53,49,42,48,63);(1,2,3,4)(5,8)(6,15)(7,22)(9,27)(10,28)(11,40)(12,29)(13,30)(14,42)(16,47)(17,48)(18,43)(19,50)(20,53)(21,59)(23,35)(24,61)(25,33)(26,39)(31,45)(32,64)(34,62)(36,58)(37,55)(38,60)(41,49)(44,66)(46,65)(51,67)(52,56)(54,68)(57,63);(5,9,32,61,12,35,10,33,13,36,21,26,11,34,38,22)(6,16,51,42,18,52,17,31,19,53,37,65,41,63,54,66)(7,8,27,64,24,29,23,28,25,30,58,59,39,40,62,60)(14,43,56,48,45,50,20,55,46,49,57,68,44,15,47,67)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=ehi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410471);