#105240

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

285 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=dgi,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, 12436);