#6573647

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4516 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)(7,19)(8,15)(9,14)(10,18)(11,17)(12,22)(13,21)(16,29)(20,36)(23,33)(24,35)(25,30)(26,32)(27,31)(28,34);(5,7,11,23)(6,12,17,32)(8,9,21,27)(10,26,29,22)(13,34,15,36)(14,30,31,35)(16,33,18,19)(20,24,28,25);(2,4)(5,8)(6,13)(7,20)(9,26)(10,24)(11,21)(12,14)(15,17)(16,35)(18,30)(19,34)(22,27)(23,28)(25,29)(31,32)(33,36);(5,9,11,27)(6,14,17,31)(7,21,23,8)(10,20,29,28)(12,30,32,35)(13,33,15,19)(16,34,18,36)(22,25,26,24);(5,8)(6,15)(7,20)(9,26)(10,24)(11,21)(12,31)(13,17)(14,32)(16,30)(18,35)(19,36)(22,27)(23,28)(25,29)(33,34)

Polycyclic presentation:

2^9|b^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480843);