#499714

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,4)(9,10)(11,20)(12,16)(13,28)(14,19)(15,21)(17,25)(18,26)(22,39)(23,36)(24,35)(27,33)(29,34)(30,40)(31,38)(32,37);(5,6)(7,8)(9,11,14,23)(10,15,19,31)(12,24,27,40)(13,29,26,22)(16,20,33,36)(17,34,32,39)(18,35,28,30)(21,37,38,25);(2,4)(6,8);(10,16)(12,25)(13,26)(15,20)(17,18)(19,33)(21,24)(22,29)(27,37)(28,32)(30,39)(31,36)(34,35)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406910);