#6574065

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
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

2500 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)(5,6,11,16)(7,13,19,28)(8,14,23,34)(9,24,10,15)(12,25,29,18)(17,30,35,36)(20,31,21,27)(22,33,26,32);(5,7,10,25,11,19,9,18)(6,12,24,28,16,29,15,13)(8,17,26,20,23,35,22,21)(14,27,33,36,34,31,32,30);(2,4)(5,7)(6,13)(8,20)(9,18)(10,25)(11,19)(12,24)(14,31)(15,29)(16,28)(17,22)(21,23)(26,35)(27,34)(30,33)(32,36);(5,8)(6,14)(7,17)(9,22)(10,26)(11,23)(12,27)(13,30)(15,32)(16,34)(18,21)(19,35)(20,25)(24,33)(28,36)(29,31);(5,7)(6,13)(8,21)(9,18)(10,25)(11,19)(12,24)(14,27)(15,29)(16,28)(17,26)(20,23)(22,35)(30,32)(31,34)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6481261);