#122944

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₈	C₂²×C₁₆
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₈×D₈
prime radical	C₂×D₈	C₂×C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

161 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30140);