#512103

order	512
nilpotency class	4
isoclinism	isoclinic class of #511753
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

969 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 969 conjugacy classes of subgroups — too many to draw. Only its 162 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)(9,10,13,19)(11,23,24,40)(12,26,28,39)(14,31,15,33)(16,37,30,22)(17,35,38,25)(18,34,27,29)(20,36,21,32);(2,5)(6,8)(9,10)(11,17)(12,27)(13,19)(14,20)(15,21)(16,22)(18,28)(23,25)(24,38)(26,34)(29,39)(30,37)(31,32)(33,36)(35,40);(1,3,4,7)(2,6,5,8)(9,11,13,24)(10,17,19,38)(12,16,28,30)(14,29,15,34)(18,37,27,22)(20,39,21,26)(23,33,40,31)(25,36,35,32);(9,12)(10,18)(11,16)(13,28)(14,25)(15,35)(17,37)(19,27)(20,40)(21,23)(22,38)(24,30)(26,33)(29,36)(31,39)(32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419299);