#122960

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

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

155 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 155 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

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, 30156);