#6573984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,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, 6481180);