#6444156

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351352);