#133384

order	512
nilpotency class	4
isoclinism	isoclinic class of #133374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#535 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 857 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=eh,f^b=fg,d^c=dh,e^c=ei,f^c=fi,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, 40580);