#103760

order	512
nilpotency class	3
isoclinism	isoclinic class of #103653
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

471 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,7)(4,6)(5,8)(9,10,15,12)(11,19,23,20)(13,18,14,17)(16,22,24,21)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(2,6)(3,4)(7,8)(9,11,15,23)(10,16,12,24)(13,22,14,21)(17,20,18,19);(1,3,5,4)(2,7,8,6)(9,12,15,10)(11,20,23,19)(13,17,14,18)(16,21,24,22)(25,27,29,35)(26,30,32,39)(28,37,38,34)(31,40,36,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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, 10956);