#151090

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂×D₄	C₂×C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 141 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) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58286);