#149726

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂²×D₄	C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 268 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10,12,16,13,17,24,28,14,18,25,29,26,30,37,38,15,19,11,20,21,31,22,33,27,32,23,34,35,39,36,40);(2,5)(3,7);(9,11,13,22,14,23,26,36,15,12,21,24,27,25,35,37)(10,16,17,28,18,29,30,38,19,20,31,33,32,34,39,40)

Polycyclic presentation:

2^9|a^2=e,c^2=fg,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,e^b=ef

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, 56922);