#6574220

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2619 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)(9,10,15,19)(11,16,22,31)(12,23,25,33)(13,18,14,27)(17,28,34,24)(20,35,37,40)(21,30,26,32)(29,36,39,38);(1,2,3,4)(5,6)(7,8)(9,11)(10,16)(12,20)(13,26)(14,21)(15,22)(17,29)(18,32)(19,31)(23,35)(24,38)(25,37)(27,30)(28,36)(33,40)(34,39);(6,8)(9,12,15,25)(10,17,19,34)(11,20,22,37)(13,24,14,28)(16,29,31,39)(18,33,27,23)(21,36,26,38)(30,35,32,40);(5,6)(7,8)(9,13,15,14)(10,18,19,27)(11,21,22,26)(12,24,25,28)(16,30,31,32)(17,33,34,23)(20,36,37,38)(29,35,39,40);(6,8)(11,22)(16,31)(20,37)(21,26)(29,39)(30,32)(35,40)(36,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481416);