#6574084

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
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

2553 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)(7,16)(8,13)(9,24)(10,14)(11,15)(12,19)(17,31)(18,20)(21,26)(22,29)(23,30)(25,28)(27,33)(32,34)(35,36);(5,7,9,19,11,20,10,25)(6,12,14,16,15,28,24,18)(8,17,22,33,23,35,26,34)(13,27,21,31,30,32,29,36);(2,4)(5,6)(7,16)(8,21)(9,14)(10,24)(11,15)(12,25)(13,26)(17,27)(18,20)(19,28)(22,30)(23,29)(31,33)(32,35)(34,36);(5,8)(6,13)(7,17)(9,22)(10,26)(11,23)(12,27)(14,21)(15,30)(16,31)(18,36)(19,33)(20,35)(24,29)(25,34)(28,32);(5,6)(7,18)(8,21)(9,14)(10,24)(11,15)(12,19)(13,26)(16,20)(17,32)(22,30)(23,29)(25,28)(27,35)(31,34)(33,36)

Polycyclic presentation:

2^9|b^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,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, 6481280);