Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#547 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,8)(13,14,19,16)(15,23,25,20)(17,21,24,26)(18,22,28,27);(10,12)(13,15,19,25)(14,20,16,23)(17,18,24,28)(21,27,26,22);(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,16,19,14)(15,20,25,23)(17,26,24,21)(18,27,28,22);(5,6)(7,8)(13,17)(14,21)(15,18)(16,26)(19,24)(20,27)(22,23)(25,28);(1,2)(3,4)(5,6)(7,8)(13,18)(14,22)(15,24)(16,27)(17,25)(19,28)(20,21)(23,26)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,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, 1167248);