#6574230

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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

2525 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)(7,12)(8,22)(9,14)(10,16)(11,17)(13,24)(15,29)(18,31)(19,20)(21,27)(23,34)(25,32)(26,36)(28,33)(30,35);(1,2,3,4)(5,7)(6,12)(8,23)(9,11)(10,20)(13,31)(14,17)(15,27)(16,19)(18,24)(21,29)(22,34)(25,35)(26,28)(30,32)(33,36);(5,8,11,25)(6,13,17,33)(7,18,9,26)(10,28,29,24)(12,30,14,34)(15,32,16,22)(19,31,27,36)(20,23,21,35);(5,9,11,7)(6,14,17,12)(8,23,25,35)(10,21,29,20)(13,31,33,36)(15,19,16,27)(18,28,26,24)(22,34,32,30);(5,6)(7,19)(8,13)(9,27)(10,15)(11,17)(12,20)(14,21)(16,29)(18,31)(22,24)(23,30)(25,33)(26,36)(28,32)(34,35)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6481426);