#102784

order	512
nilpotency class	3
isoclinism	isoclinic class of #102749
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

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

405 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9980);