Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#39106 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5366 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);(4,5)(8,9)(20,24)(25,26);(3,4,7,5)(6,9,10,8)(12,15)(13,17);(3,4,7,5)(6,8,10,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,26)(23,24);(3,5,7,4)(6,9,10,8)(19,21,23,22)(20,25,24,26);(3,6)(4,8)(5,9)(7,10)(11,13,14,17)(12,16,15,18)(19,22,23,21)(20,26,24,25)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543624);