#6573653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4423 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)(7,16)(8,22)(9,26)(10,14)(11,15)(12,19)(13,25)(17,21)(18,32)(20,29)(23,24)(27,30)(28,35)(31,34)(33,36);(5,7,11,21)(6,12,15,29)(8,18,25,34)(9,20,10,19)(13,28,22,36)(14,17,26,16)(23,32,30,31)(24,35,27,33);(2,4)(5,6)(7,17)(8,23)(9,14)(10,26)(11,15)(12,19)(13,27)(16,21)(18,28)(20,29)(22,24)(25,30)(31,33)(32,35)(34,36);(5,8)(6,13)(7,18)(9,24)(10,27)(11,25)(12,28)(14,23)(15,22)(16,31)(17,32)(19,33)(20,35)(21,34)(26,30)(29,36);(5,6)(7,17)(8,23)(9,14)(10,26)(11,15)(12,19)(13,27)(16,21)(18,28)(20,29)(22,24)(25,30)(31,33)(32,35)(34,36)

Polycyclic presentation:

2^9|b^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,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, 6480849);