#6573650

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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

4589 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)(8,13)(9,11)(10,16)(12,20)(15,25)(17,26)(18,34)(19,28)(21,31)(22,27)(23,33)(24,32)(29,36)(30,35);(6,11)(7,17)(9,25)(13,27)(15,16)(18,21)(19,20)(22,24)(26,28)(30,31)(32,33)(35,36);(2,4)(5,7,10,20)(6,12,16,14)(8,21,23,35)(9,19,25,17)(11,26,15,28)(13,30,33,31)(18,24,36,22)(27,29,32,34);(5,8,10,23)(6,13,16,33)(7,18,20,36)(9,22,25,24)(11,27,15,32)(12,29,14,34)(17,21,19,35)(26,31,28,30);(5,7,10,20)(6,14,16,12)(8,21,23,35)(9,19,25,17)(11,28,15,26)(13,31,33,30)(18,24,36,22)(27,34,32,29)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^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=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480846);