Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C231 |
| direct product | C23×#4928938 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C25 |
| commutator, Frattini, cosocle | C24 | C28 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
129585515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 8 generators on 26 points:
(11,12)(13,14)(19,20)(21,24)(22,23)(25,26);(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(20,23)(21,25);(3,4);(5,6);(15,16)(17,18)(21,25)(24,26);(1,2)(3,4)(5,6);(8,9)(12,13)(16,17)(19,20)(21,26)(22,23)(24,25);(7,9,10,8)(11,13,14,12)(15,17,18,16)(19,21,22,25)(20,24,23,26)832 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((11,12)(13,14)(19,20)(21,24)(22,23)(25,26), (7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(20,23)(21,25), (3,4), (5,6), (15,16)(17,18)(21,25)(24,26), (1,2)(3,4)(5,6), (8,9)(12,13)(16,17)(19,20)(21,26)(22,23)(24,25), (7,9,10,8)(11,13,14,12)(15,17,18,16)(19,21,22,25)(20,24,23,26));