#358724

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂³
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=ef,h^2=i,b^a=be,c^a=cf,c^b=cg

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, 265920);