#491679

order	512
nilpotency class	4
isoclinism	isoclinic class of #491644
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 ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

574 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 150 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)(19,28)(20,26)(21,24)(22,36)(23,37)(25,32)(27,29)(30,42)(31,41)(33,47)(34,39)(35,48)(38,46)(40,45)(43,44);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(18,24)(25,38)(26,37)(27,39)(28,41)(36,48)(40,43)(42,47);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16)(17,19,29,44,23,33,35,46,21,31,34,45,20,30,22,32)(18,25,36,42,26,40,39,41,24,38,48,47,37,43,27,28);(3,10)(8,15)(11,14)(12,16)(17,20,21,23)(18,26,24,37)(19,30,31,33)(22,34,35,29)(25,40,38,43)(27,36,39,48)(28,42,41,47)(32,45,46,44)

Polycyclic presentation:

2^9|b^2=egi,c^2=fh,d^2=i,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^c=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398875);