#491571

order	512
nilpotency class	4
isoclinism	isoclinic class of #491565
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 613 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fhi,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,c^b=ci,d^b=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, 398767);