#491103

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

467 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 128 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)(10,15)(12,16)(13,24)(17,26)(18,34)(19,29)(20,35)(22,30)(23,38)(25,31)(27,28)(32,37)(33,40)(36,39);(9,10,16,28,13,18,25,29,14,19,31,34,24,27,12,15)(11,20,30,40,23,36,37,26,21,17,32,39,38,33,22,35);(2,5)(3,8)(6,7)(9,11)(10,17)(12,22)(13,23)(14,21)(15,26)(16,30)(18,33)(19,20)(24,38)(25,37)(27,36)(28,39)(29,35)(31,32)(34,40);(2,6)(5,7)(11,21)(17,20)(22,37)(23,38)(26,35)(30,32)(33,36)(39,40)

Polycyclic presentation:

2^9|b^2=eg,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, 398299);