Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,8)(14,18)(15,21)(16,23)(27,28);(10,12)(13,14,17,18)(15,19,21,25)(16,20,23,26)(22,27,24,28);(1,2,4,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,25)(22,27)(23,26)(24,28);(5,6)(7,8)(13,15,17,21)(14,19,18,25)(16,24,23,22)(20,28,26,27);(1,3,4,2)(5,6)(7,8)(13,16,17,23)(14,20,18,26)(15,22,21,24)(19,27,25,28)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=f,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1284519);