Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,8,10)(7,11,12,9)(15,19)(17,20)(21,22)(23,30)(24,34)(25,29)(26,27)(28,33)(31,32)(35,36);(5,6,8,10)(7,9,12,11)(13,14,16,18)(15,17,19,20)(22,27)(24,29)(28,30)(31,35);(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,33)(22,28,27,30)(24,31,29,35)(25,32,34,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(21,24)(22,25)(23,31)(26,29)(27,34)(28,32)(30,36)(33,35);(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20)(21,25)(22,29)(23,32)(24,27)(26,34)(28,35)(30,31)(33,36)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4774322);