#104034

order	512
nilpotency class	3
isoclinism	isoclinic class of #103989
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

424 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11230);