Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
| direct product | C2×#49876 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2456 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5906 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);(11,12,15,17)(13,14,18,16)(21,25)(24,26);(4,7)(5,9)(11,13)(12,16)(14,17)(15,18)(19,20,23,22)(21,24,25,26);(3,4,6,7)(5,8,9,10)(11,13,15,18)(12,16,17,14)(19,21)(20,24)(22,26)(23,25);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,18)(16,17);(3,5)(4,8)(6,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,22,23,20)(21,26,25,24)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612627);