#146965

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54161);