#507619

order	512
nilpotency class	4
isoclinism	isoclinic class of #500891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eg,c^2=g,d^2=ei,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fhi,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, 414815);