#6574072

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
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

2521 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)(5,6,11,16)(7,13,19,27)(8,14,22,32)(9,23,10,15)(12,24,28,18)(17,29,34,36)(20,26,33,35)(21,31,25,30);(5,7,10,24,11,19,9,18)(6,12,23,27,16,28,15,13)(8,17,25,33,22,34,21,20)(14,26,31,36,32,35,30,29);(2,4)(5,7)(6,13)(8,20)(9,18)(10,24)(11,19)(12,23)(14,26)(15,28)(16,27)(17,25)(21,34)(22,33)(29,30)(31,36)(32,35);(5,8)(6,14)(7,17)(9,21)(10,25)(11,22)(12,26)(13,29)(15,30)(16,32)(18,20)(19,34)(23,31)(24,33)(27,36)(28,35);(5,7)(6,13)(8,20)(9,18)(10,24)(11,19)(12,23)(14,26)(15,28)(16,27)(17,25)(21,34)(22,33)(29,30)(31,36)(32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481268);