#133506

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

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

normal subgroups only

This group has 630 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)(9,10,14,20,15,21,13,19)(11,22,24,33,25,35,29,17)(12,16,28,32,26,18,27,31)(23,36,39,34,38,37,40,30);(1,2)(3,5)(4,6)(7,8)(10,16)(12,26)(13,14)(17,30)(18,21)(19,31)(20,32)(22,36)(23,38)(24,29)(33,34)(35,37);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17,21,33)(12,23)(13,24)(14,29)(15,25)(16,30,18,34)(19,22,20,35)(26,38)(27,39)(28,40)(31,36,32,37)

Polycyclic presentation:

2^9|a^2=f,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, 40702);