#6574172

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^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, 6481368);