#129235

order	512
nilpotency class	4
isoclinism	isoclinic class of #12711
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

1466 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,f^b=fg,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36431);