#492517

order	512
nilpotency class	4
isoclinism	isoclinic class of #492512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

501 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,c^b=ci,d^b=de,e^b=eg,g^b=gi,f^c=fh,e^d=eg,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399713);