Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#35999 |
| 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 | |
5740 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);(12,15)(16,18)(19,20,23,22)(21,25,26,24);(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,18)(19,21,23,26)(20,24,22,25);(3,5)(4,7)(6,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,20,23,22)(21,24,26,25);(3,5)(4,8)(6,10)(7,9)(11,13,14,17)(12,16,15,18)(19,22,23,20)(21,25,26,24);(3,4)(5,7)(6,9)(8,10)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536488);