#104909

order	512
nilpotency class	3
isoclinism	isoclinic class of #104897
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#394
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

305 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=d,d^2=g,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,f^b=fhi,e^c=egh,f^c=fghi

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, 12105);