#495434

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,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, 402630);