#6573989

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

2517 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)(8,20)(9,26)(10,14)(11,15)(13,23)(16,32)(17,27)(18,29)(19,30)(21,28)(22,36)(24,33)(25,34)(31,35);(1,2,3,4)(5,7,11,19)(6,12,15,30)(8,21,25,22)(9,18,10,27)(13,31,33,32)(14,17,26,29)(16,23,35,24)(20,28,34,36);(5,8,11,25)(6,13,15,33)(7,16,19,35)(9,24,10,23)(12,28,30,36)(14,34,26,20)(17,31,29,32)(18,22,27,21);(5,7,11,19)(6,12,15,30)(8,22,25,21)(9,18,10,27)(13,32,33,31)(14,17,26,29)(16,24,35,23)(20,36,34,28);(5,6)(7,17)(8,13)(9,14)(10,26)(11,15)(12,27)(16,31)(18,30)(19,29)(20,23)(21,28)(22,36)(24,34)(25,33)(32,35)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=i,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, 6481185);