#501215

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^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, 408411);