#6574078

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

2511 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,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, 6481274);