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×#36088 |
| 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 | |
5546 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);(3,4,9,10)(5,6,8,7)(11,12,15,18)(13,16,14,17)(19,20,22,24)(21,23,25,26);(3,5,9,8)(4,7,10,6)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26);(3,6,9,7)(4,5,10,8)(11,12)(13,16)(14,17)(15,18);(3,7,9,6)(4,5,10,8)(11,13,15,14)(12,16,18,17);(3,8,9,5)(4,7,10,6)(11,14,15,13)(12,17,18,16)(19,21,22,25)(20,23,24,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536793);