#505582

order	512
nilpotency class	4
isoclinism	isoclinic class of #500109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 627 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)(3,7)(4,6)(5,8)(9,10,12,17,14,19,15,29)(11,20,30,39,21,37,22,36)(13,25,24,35,26,38,28,18)(16,31,33,23,34,27,32,40);(1,3,4,8)(2,5,6,7)(11,21)(13,26)(18,35)(20,37)(22,30)(24,28)(25,38)(36,39);(9,11)(10,16)(12,22)(13,27)(14,21)(15,30)(17,32)(18,36)(19,34)(20,38)(23,24)(25,37)(26,31)(28,40)(29,33)(35,39);(2,6)(5,7)(9,12,14,15)(10,17,19,29)(11,22,21,30)(13,24,26,28)(16,32,34,33)(18,25,35,38)(20,36,37,39)(23,31,40,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=e,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,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, 412778);