#489388

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

781 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=gi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

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, 396584);