#102780

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	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 447 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,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, 9976);