#392804

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1934 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300000);