#133389

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, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 794 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,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, 40585);