#512107

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

982 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 164 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)(3,6)(4,5)(7,8)(9,10,15,21)(11,22,26,38)(12,16,30,19)(13,31,27,18)(14,32,29,20)(17,28,37,24)(23,36,40,39)(25,35,33,34);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,16)(12,27)(13,30)(14,29)(17,34)(18,20)(19,21)(22,36)(23,24)(25,33)(28,40)(31,32)(35,37)(38,39);(9,11)(10,17)(12,28)(13,23)(14,33)(15,26)(16,34)(18,38)(19,35)(20,39)(21,37)(22,31)(24,30)(25,29)(27,40)(32,36)

Polycyclic presentation:

2^9|a^2=i,c^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,h^a=hi,f^c=fh,g^c=ghi,h^c=hi,f^d=fhi,g^d=ghi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419303);