#103967

order	512
nilpotency class	3
isoclinism	isoclinic class of #103845
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

448 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11163);