#493220

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

808 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)(9,10,13,18,14,12,24,25,15,19,27,33,28,26,17,32)(11,16,22,31,23,21,37,38,20,29,35,39,36,34,30,40);(2,4)(5,6,8,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,20)(17,30)(18,31)(19,29)(24,37)(25,38)(26,34)(27,35)(28,36)(32,40)(33,39);(5,7,8,6)(11,20)(16,29)(21,34)(22,35)(23,36)(30,37)(31,39)(38,40);(9,12,13,25,14,19,24,33,15,26,27,32,28,10,17,18)(11,21,22,38,23,29,37,39,20,34,35,40,36,16,30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,h^2=i,b^a=be,d^a=di,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400416);