#133462

order	512
nilpotency class	4
isoclinism	isoclinic class of #133338
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, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 638 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,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, 40658);