#133413

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	C₂³×D₄	C₂³
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 897 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)(11,19)(12,23,26,16)(13,27,28,17)(14,18)(15,21,30,37)(20,34,24,38)(22,35)(25,32)(29,36)(31,39)(33,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,22)(10,15,18,30)(12,24,26,20)(13,21,28,37)(16,31,23,29)(17,19,27,35)(25,39,40,36)(32,34,33,38);(9,12,14,26)(10,16,18,23)(11,20,22,24)(13,25,28,40)(15,29,30,31)(17,33,27,32)(19,34,35,38)(21,36,37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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,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, 40609);