#6574066

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
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

2534 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)(7,13)(8,21)(9,14)(10,17)(11,16)(12,19)(15,22)(18,30)(20,29)(23,36)(24,31)(25,34)(26,33)(27,32)(28,35);(5,7,10,19,11,8,22,20)(6,12,15,13,16,29,17,21)(9,18,25,35,26,23,27,24)(14,28,32,30,33,31,34,36);(2,4)(5,8)(6,13)(7,11)(9,24)(10,20)(12,17)(14,28)(15,29)(16,21)(18,25)(19,22)(23,27)(26,35)(30,32)(31,33)(34,36);(5,9,11,26)(6,14,16,33)(7,18,8,23)(10,25,22,27)(12,28,29,31)(13,30,21,36)(15,32,17,34)(19,35,20,24);(5,7)(6,13)(8,11)(9,24)(10,19)(12,17)(14,31)(15,29)(16,21)(18,25)(20,22)(23,27)(26,35)(28,33)(30,34)(32,36)

Polycyclic presentation:

2^9|b^2=f,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,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, 6481262);