#491104

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 130 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)(4,6)(7,8)(10,16)(11,22)(12,14)(13,17)(18,34)(19,31)(20,30)(21,32)(23,38)(24,26)(25,40)(27,28)(29,37)(36,39);(9,10,17,30,12,19,28,32,15,21,27,31,14,20,13,16)(11,23,33,38,22,36,40,29,26,18,35,34,24,37,25,39);(2,5)(3,8)(6,7)(9,11)(10,18)(12,22)(13,25)(14,24)(15,26)(16,29)(17,33)(19,37)(20,36)(21,23)(27,35)(28,40)(30,34)(31,38)(32,39);(2,6)(5,7)(9,12,15,14)(10,19,21,20)(11,24,26,22)(13,17,28,27)(16,30,32,31)(18,36,23,37)(25,35,40,33)(29,38,39,34)

Polycyclic presentation:

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

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, 398300);