#105224

order	512
nilpotency class	3
isoclinism	isoclinic class of #105209
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#397
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

248 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12420);