#135355

order	512
nilpotency class	4
isoclinism	isoclinic class of #124654
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

876 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=d,e^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^c=eh,f^c=fgh,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42551);