#6574085

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
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

2507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481281);