#181095

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1711 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,g^2=hi,b^a=be,e^a=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88291);