#103780

order	512
nilpotency class	3
isoclinism	isoclinic class of #103653
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

453 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,c^b=cf,d^b=dg,f^b=fh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10976);