Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35560 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2428 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6700 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);(3,4,7,5)(6,8,10,9)(11,12)(13,15)(14,17)(16,18);(3,4,7,5)(6,9,10,8)(11,13,16,14)(12,17,18,15)(19,20)(21,22)(23,24)(25,26);(4,5)(8,9)(12,18)(13,14)(20,24)(21,25);(3,5,7,4)(6,9,10,8)(11,14,16,13)(12,17,18,15)(19,21,23,25)(20,22,24,26);(3,6)(4,8)(5,9)(7,10)(11,15,16,17)(12,13,18,14)(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, 7536137);