#96708

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

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

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, 3904);