#133417

order	512
nilpotency class	4
isoclinism	isoclinic class of #133406
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, cosocle	C₂²×#41	C₂³
prime radical	#2431	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 818 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,e^b=ehi,f^b=fg,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, 40613);