Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19142 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C23 | C2×#121 |
| prime radical | C2×#3100 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1998 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)(5,7)(6,8)(9,10)(11,12)(13,20)(14,18)(15,24)(16,19)(17,23)(21,26)(22,25);(4,7)(8,9)(11,13,15,23,16,21,14,22)(12,17,18,20,19,25,24,26);(4,8)(6,10)(7,9)(13,21)(17,25)(20,26)(22,23);(11,14,16,15)(12,18,19,24)(13,22,21,23)(17,20,25,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260806);