#6574080

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574077
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

2582 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)(9,26)(10,14)(11,15)(12,19)(17,33)(18,20)(21,36)(22,24)(23,28)(25,32)(27,30)(29,34)(31,35);(5,7,10,27,11,20,9,19)(6,12,26,18,15,30,14,16)(8,21,28,34,25,17,24,35)(13,31,22,33,32,29,23,36);(2,4)(5,6)(7,16)(8,22)(9,14)(10,26)(11,15)(12,27)(13,24)(17,31)(18,20)(19,30)(21,29)(23,25)(28,32)(33,35)(34,36);(5,8)(6,13)(7,17)(9,24)(10,28)(11,25)(12,29)(14,23)(15,32)(16,33)(18,36)(19,34)(20,21)(22,26)(27,35)(30,31);(5,6)(7,18)(8,23)(9,14)(10,26)(11,15)(12,19)(13,28)(16,20)(17,31)(21,29)(22,25)(24,32)(27,30)(33,35)(34,36)

Polycyclic presentation:

2^9|b^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,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, 6481276);