#105239

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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=di,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, 12435);