#512273

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fh,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^c=gi,f^d=fi,g^d=ghi,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, 419469);