#102778

order	512
nilpotency class	3
isoclinism	isoclinic class of #102749
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

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

normal subgroups only

This group has 451 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 9974);