#6573638

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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

4430 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)(8,13)(9,26)(10,14)(11,15)(12,19)(17,21)(18,31)(20,29)(22,24)(23,27)(25,30)(28,34)(32,35)(33,36);(5,7,11,21)(6,12,15,29)(8,18,25,35)(9,20,10,19)(13,28,30,33)(14,17,26,16)(22,31,23,32)(24,36,27,34);(2,4)(5,6)(7,17)(8,22)(9,14)(10,26)(11,15)(12,19)(13,24)(16,21)(18,33)(20,29)(23,25)(27,30)(28,35)(31,36)(32,34);(5,8,11,25)(6,13,15,30)(7,18,21,35)(9,24,10,27)(12,28,29,33)(14,23,26,22)(16,31,17,32)(19,34,20,36);(5,6)(7,17)(8,23)(9,14)(10,26)(11,15)(12,19)(13,27)(16,21)(18,28)(20,29)(22,25)(24,30)(31,34)(32,36)(33,35)

Polycyclic presentation:

2^9|b^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,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, 6480834);