#133355

order	512
nilpotency class	4
isoclinism	isoclinic class of #133350
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

795 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 795 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,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, 40551);