#151660

order	512
nilpotency class	5
isoclinism	isoclinic class of #151637
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂×D₄	D₁₆
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

138 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=df,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,h^a=hi,e^c=eg

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58856);