Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| direct product | C2×#72334357 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C27 |
| commutator, Frattini, cosocle | C24 | C28 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
At least 16428 conjugacy classes of subgroups; only the 523627 normal subgroups are shown.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 26 points:
(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25);(11,13)(12,15)(14,17)(16,18)(19,21)(20,24)(22,26)(23,25);(4,6)(8,10)(11,14)(12,13)(15,16)(17,18);(4,6)(12,17)(14,15);(8,10)(19,21)(20,22)(23,25)(24,26);(1,2);(11,13)(12,14)(15,17)(16,18);(3,4)(5,6)(7,8)(9,10)(11,15,16,14)(12,18,17,13)(19,22)(20,25)(21,26)(23,24)592 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25), (11,13)(12,15)(14,17)(16,18)(19,21)(20,24)(22,26)(23,25), (4,6)(8,10)(11,14)(12,13)(15,16)(17,18), (4,6)(12,17)(14,15), (8,10)(19,21)(20,22)(23,25)(24,26), (1,2), (11,13)(12,14)(15,17)(16,18), (3,4)(5,6)(7,8)(9,10)(11,15,16,14)(12,18,17,13)(19,22)(20,25)(21,26)(23,24));