#490723

order	512
nilpotency class	4
isoclinism	isoclinic class of #490699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

558 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=eg,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=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397919);