Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#1112089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C24 | C26 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
30647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(4,6)(8,10)(12,15)(13,17)(19,20,22,24)(21,25,26,23);(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21,22,26)(20,23,24,25);(12,15)(16,18)(19,20,22,24)(21,25,26,23);(1,2);(3,4,5,6)(11,12)(13,18)(14,15)(16,17)(19,20)(21,23)(22,24)(25,26);(11,13)(12,16)(14,17)(15,18)(19,21)(20,23)(22,26)(24,25)172 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,6)(8,10)(12,15)(13,17)(19,20,22,24)(21,25,26,23), (7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21,22,26)(20,23,24,25), (12,15)(16,18)(19,20,22,24)(21,25,26,23), (1,2), (3,4,5,6)(11,12)(13,18)(14,15)(16,17)(19,20)(21,23)(22,24)(25,26), (11,13)(12,16)(14,17)(15,18)(19,21)(20,23)(22,26)(24,25));