#499631

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,g^d=gi

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, 406827);