#393205

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,c^b=ci,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300401);