#140615

order	512
nilpotency class	4
isoclinism	isoclinic class of #13229
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

545 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,f^b=fg,g^b=gi,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, 47811);