#491690

order	512
nilpotency class	4
isoclinism	isoclinic class of #491662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

566 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18,23,27)(19,24,29,39)(20,32,35,36)(21,34,37,41)(22,33,25,26)(28,42,43,47)(30,45,46,48)(31,44,38,40);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,38)(26,40)(27,39)(32,42)(33,44)(34,45)(35,43)(36,47)(37,46)(41,48);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,20,32,27,22,34,21,33,23,35,36,18,25,41,37,26)(19,28,42,39,31,45,30,44,29,43,47,24,38,48,46,40);(3,10)(8,15)(11,14)(12,16)(19,29)(24,39)(28,43)(30,46)(31,38)(40,44)(42,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398886);