#133313

order	512
nilpotency class	4
isoclinism	isoclinic class of #133281
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	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

856 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 84 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)(9,10,13,18,14,17,28,34)(11,19,21,35,22,15,29,31)(12,23,25,36,26,16,27,33)(20,30,37,39,24,32,38,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,22)(10,15,17,19)(12,24,26,20)(13,21,28,29)(16,32,23,30)(18,35,34,31)(25,38,27,37)(33,39,36,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,26)(10,16,17,23)(11,20,22,24)(13,27,28,25)(15,30,19,32)(18,36,34,33)(21,38,29,37)(31,40,35,39)

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,c^b=ci,e^b=eh,f^b=fg,d^c=dh,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, 40509);