Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#59677 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,10)(7,8)(12,17)(15,18)(19,20,23,24)(21,26,25,22);(4,8)(5,10)(11,12,16,17)(13,18,14,15)(20,24)(21,25);(3,5)(4,6)(7,10)(8,9)(11,13,16,14)(12,18,17,15)(19,21,23,25)(20,22,24,26);(3,5)(4,6)(7,10)(8,9)(11,14,16,13)(12,15,17,18)(19,22)(20,25)(21,24)(23,26);(3,6)(4,5)(7,9)(8,10)(11,15)(12,13)(14,17)(16,18)(19,22)(20,25)(21,24)(23,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573137);