#128508

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

294 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,e^2=h,f^2=g,h^2=i,b^a=bd,f^a=fg,c^b=ci,d^b=dg

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, 35704);