#392796

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1611 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)(7,8)(9,10,13,17,14,12,21,16)(11,15,19,23,20,18,24,22);(1,2,4,3)(9,11,13,19,14,20,21,24)(10,15,17,23,12,18,16,22);(1,3,4,2)(9,12)(10,14)(11,18)(13,16)(15,20)(17,21)(19,22)(23,24);(6,8)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299992);