#6574073

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

2555 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)(7,13)(8,21)(9,14)(10,17)(11,16)(12,19)(15,22)(18,31)(20,30)(23,36)(24,29)(25,32)(26,35)(27,34)(28,33);(5,7,10,19,11,8,22,20)(6,12,15,13,16,30,17,21)(9,18,26,24,27,23,28,25)(14,29,33,31,34,32,35,36);(2,4)(5,8)(6,13)(7,11)(9,24)(10,20)(12,17)(14,32)(15,30)(16,21)(18,28)(19,22)(23,26)(25,27)(29,34)(31,35)(33,36);(5,9,11,27)(6,14,16,34)(7,18,8,23)(10,26,22,28)(12,29,30,32)(13,31,21,36)(15,33,17,35)(19,24,20,25);(5,7)(6,13)(8,11)(9,25)(10,19)(12,17)(14,32)(15,30)(16,21)(18,26)(20,22)(23,28)(24,27)(29,34)(31,35)(33,36)

Polycyclic presentation:

2^9|b^2=f,d^2=i,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, 6481269);