#122475

order	512
nilpotency class	4
isoclinism	isoclinic class of #11663
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	C₂³	D₄⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

273 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(2,6,7,14)(3,9)(4,12)(8,10,13,15)(11,16)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(1,2,5,7)(3,10,11,15)(4,13,12,8)(6,9,14,16);(18,22)(23,28)(24,29)(30,32)

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh

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, 29671);