#128249

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₁₆	D₄
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	C₂×#41	C₄²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

410 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35445);