#490478

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

528 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 148 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)(9,10)(11,20)(12,17)(13,27)(14,26)(15,19)(16,23)(18,28)(21,33)(22,34)(24,40)(25,36)(29,30)(31,37)(32,39)(35,38);(9,11,21,37,14,24,13,23,15,25,28,38,12,22,29,39)(10,16,30,40,17,31,18,20,19,32,27,34,26,35,33,36);(2,5)(3,8)(6,7)(9,10)(11,16)(12,26)(13,18)(14,17)(15,19)(20,23)(21,30)(22,35)(24,31)(25,32)(27,28)(29,33)(34,38)(36,39)(37,40);(9,12,15,14)(10,17,19,26)(11,22,25,24)(13,21,29,28)(16,31,32,35)(18,27,33,30)(20,34,36,40)(23,37,39,38)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397674);