#6574104

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574100
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2543 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:

(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,21)(13,23)(18,32)(19,27)(20,28)(24,29)(25,30)(26,34)(31,35)(33,36);(1,2)(3,4)(5,7,9,21,11,20,10,19)(6,12,15,17,16,27,14,28)(8,18,23,35,24,34,25,33)(13,26,22,31,30,32,29,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,34)(21,35)(27,32)(28,36);(1,2)(3,4)(5,9,11,10)(6,14,16,15)(7,19,20,21)(8,23,24,25)(12,17,27,28)(13,29,30,22)(18,33,34,35)(26,31,32,36);(2,4)(7,20)(12,27)(17,28)(18,34)(19,21)(26,32)(31,36)(33,35)

Polycyclic presentation:

2^9|b^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

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, 6481300);