#6573658

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

4496 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)(7,14)(8,21)(9,11)(10,16)(12,20)(13,24)(15,26)(17,27)(18,30)(19,29)(22,31)(23,33)(25,28)(32,35)(34,36);(6,11)(7,17)(9,26)(13,28)(15,16)(18,22)(19,20)(21,33)(23,25)(27,29)(31,32)(35,36);(2,4)(5,7,10,20)(6,12,16,14)(8,22,24,35)(9,19,26,17)(11,27,15,29)(13,31,21,32)(18,25,36,23)(28,30,33,34);(5,8)(6,13)(7,18)(9,23)(10,24)(11,28)(12,30)(14,34)(15,33)(16,21)(17,22)(19,35)(20,36)(25,26)(27,32)(29,31);(5,7,10,20)(6,14,16,12)(8,22,24,35)(9,19,26,17)(11,29,15,27)(13,32,21,31)(18,25,36,23)(28,34,33,30)

Polycyclic presentation:

2^9|c^2=i,e^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, 6480854);