#6574369

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2667 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)(11,16,23,32)(12,24,26,33)(13,18,14,27)(17,28,34,25)(20,35,38,39)(21,30,22,31)(29,36,40,37);(1,2,3,4)(5,6)(7,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,23)(17,29)(18,31)(19,32)(24,35)(25,37)(26,38)(27,30)(28,36)(33,39)(34,40);(6,8)(9,12,15,26)(10,17,19,34)(11,20,23,38)(13,25,14,28)(16,29,32,40)(18,33,27,24)(21,36,22,37)(30,35,31,39);(5,6)(7,8)(9,13,15,14)(10,18,19,27)(11,21,23,22)(12,25,26,28)(16,30,32,31)(17,33,34,24)(20,36,38,37)(29,35,40,39);(6,8)(9,13,15,14)(10,18,19,27)(11,22,23,21)(12,25,26,28)(16,31,32,30)(17,33,34,24)(20,37,38,36)(29,39,40,35)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6481565);