#403391

order	512
nilpotency class	3
isoclinism	isoclinic class of #393921
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2180 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 2180 conjugacy classes of subgroups — too many to draw. Only its 298 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=g,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310587);