#133451

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 636 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)(11,20,21,36)(12,17,25,34)(13,16)(14,19)(15,22)(18,28)(23,39,26,35)(24,33,27,32)(29,40)(30,37)(31,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,21)(10,15,19,30)(12,22,25,37)(13,26,28,23)(16,31,18,29)(17,20,34,36)(24,40,27,38)(32,35,33,39);(1,3)(2,5)(4,7)(6,8)(10,16,19,18)(11,21)(15,29,30,31)(17,33,34,32)(20,35,36,39)(22,37)(23,26)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fg,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, 40647);