#151104

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

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×D₈	C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

151 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 151 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58300);