#6574070

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574070
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

2555 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)(8,15)(9,14)(10,18)(11,17)(12,20)(16,22)(19,30)(21,29)(23,34)(24,28)(25,35)(26,33)(27,32)(31,36);(5,7,10,20,11,8,22,21)(6,12,16,13,17,29,18,15)(9,19,25,24,26,23,27,36)(14,28,32,30,33,31,35,34);(2,4)(5,8)(6,13)(7,11)(9,24)(10,21)(12,18)(14,31)(15,17)(16,29)(19,27)(20,22)(23,25)(26,36)(28,33)(30,35)(32,34);(5,9)(6,14)(7,19)(8,23)(10,25)(11,26)(12,28)(13,30)(15,34)(16,32)(17,33)(18,35)(20,24)(21,36)(22,27)(29,31);(5,8)(6,15)(7,11)(9,24)(10,21)(12,16)(13,17)(14,28)(18,29)(19,27)(20,22)(23,25)(26,36)(30,32)(31,33)(34,35)

Polycyclic presentation:

2^9|b^2=f,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=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481266);