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×#60041 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5600 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);(12,16)(13,18)(19,20,24,22)(21,23,26,25);(5,9)(8,10)(12,16)(13,18)(19,21,24,26)(20,25,22,23);(3,4,7,6)(5,8,9,10)(11,12)(13,17)(14,18)(15,16)(19,22,24,20)(21,23,26,25);(3,5)(4,8)(6,10)(7,9)(11,13,15,18)(12,17,16,14);(3,6,7,4)(5,10,9,8)(11,14,15,17)(12,18,16,13)(19,23)(20,21)(22,26)(24,25)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573480);