#133508

order	512
nilpotency class	4
isoclinism	isoclinic class of #133362
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

626 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40704);