#105234

order	512
nilpotency class	3
isoclinism	isoclinic class of #102857
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator, Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#399
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

295 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=deg,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fi,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12430);