#507973

order	512
nilpotency class	4
isoclinism	isoclinic class of #501332
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415169);