#133337

order	512
nilpotency class	4
isoclinism	isoclinic class of #133329
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×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

866 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,7,8,5)(4,6)(9,10,14,20,15,18,13,19)(11,21,24,32,25,16,23,33)(12,26,30,17,28,36,29,35)(22,37,39,34,27,40,38,31);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,18,21)(12,27,28,22)(13,23,14,24)(17,34,35,31)(19,33,20,32)(26,40,36,37)(29,39,30,38);(9,12,15,28)(10,17,18,35)(11,22,25,27)(13,30,14,29)(16,31,21,34)(19,36,20,26)(23,39,24,38)(32,37,33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,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, 40533);