#143295

order	512
nilpotency class	4
isoclinism	isoclinic class of #143209
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 376 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,g^b=gi,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50491);