#493188

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 400384);