Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
3238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,8)(13,14)(15,19)(16,18)(17,24)(20,23)(21,25)(22,27)(26,28);(2,3)(10,12)(13,15,18,23)(14,19,16,20)(17,22,25,28)(21,26,24,27);(9,10)(11,12)(13,15)(14,20)(16,19)(17,22)(18,23)(21,27)(24,26)(25,28);(2,3)(5,6)(7,8)(13,16,18,14)(15,19,23,20)(17,24,25,21)(22,26,28,27);(1,3,4,2)(13,17)(14,21)(15,22)(16,24)(18,25)(19,26)(20,27)(23,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2713142);