#6574169

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2591 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)(7,8);(1,2,3,4)(9,10)(11,20)(12,17)(13,27)(14,30)(15,19)(16,22)(18,32)(21,35)(23,33)(24,39)(25,38)(26,37)(28,40)(29,34)(31,36);(6,8)(9,11,15,13)(10,16,19,34)(12,23,26,24)(14,29,32,22)(17,35,37,40)(18,27,30,20)(21,25,28,31)(33,36,39,38);(9,12,15,26)(10,17,19,37)(11,21,13,28)(14,31,32,25)(16,33,34,39)(18,38,30,36)(20,35,27,40)(22,23,29,24);(9,13,15,11)(10,16,19,34)(12,24,26,23)(14,22,32,29)(17,35,37,40)(18,27,30,20)(21,31,28,25)(33,36,39,38)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481365);