#512108

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

Polycyclic presentation:

2^9|a^2=i,c^2=ei,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, 419304);