Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18123 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#120 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,6,9)(5,7,10,8)(16,22)(23,26);(4,7)(5,10)(8,9)(11,12)(13,16)(14,17)(15,18)(19,23)(20,22)(21,24)(25,26);(11,13)(12,16)(14,19)(15,20)(17,23)(18,22)(21,25)(24,26);(11,14,15,21)(12,17,18,24)(13,19,20,25)(16,23,22,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253558);