#491441

order	512
nilpotency class	4
isoclinism	isoclinic class of #491439
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

598 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 182 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)(3,7)(6,8)(10,15)(12,23)(13,24)(16,25)(17,26)(18,30)(19,29)(21,36)(22,35)(27,28)(31,34)(32,37)(33,40)(38,39);(9,10,16,30,13,18,12,15,14,19,23,28,24,27,25,29)(11,17,31,40,22,33,21,26,20,32,36,39,35,38,34,37);(1,2,3,5,4,6,7,8)(9,11)(10,17)(12,21)(13,22)(14,20)(15,26)(16,31)(18,33)(19,32)(23,36)(24,35)(25,34)(27,38)(28,39)(29,37)(30,40);(11,20)(17,32)(21,34)(22,35)(26,37)(31,36)(33,38)(39,40)

Polycyclic presentation:

2^9|b^2=egi,c^2=f,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398637);