#499632

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	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 443 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 406828);