#403569

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

999 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^c=fi,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, 310765);