#133469

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, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

619 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 80 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)(9,10,14,19,15,20,13,17)(11,21,24,39,22,35,29,38)(12,18,27,34,28,37,26,36)(16,30,32,23,33,40,31,25);(1,2)(3,5)(4,6)(7,8)(9,11,15,22)(10,16,20,33)(12,25,28,23)(13,29,14,24)(17,32,19,31)(18,35,37,21)(26,40,27,30)(34,39,36,38);(1,3)(2,5)(4,7)(6,8)(10,17,20,19)(11,22)(16,31,33,32)(18,36,37,34)(21,38,35,39)(23,25)(24,29)(30,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=i,f^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, 40665);