#6574090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574086
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=fi,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481286);