Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C29 · Schur multiplier (H2) = C234 |
| direct product | C23×#7652795 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C26 |
| commutator, Frattini, cosocle | C23 | C29 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
38165195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 9 generators on 26 points:
(11,12)(13,18)(14,16)(15,17);(11,13,15,16)(12,14,17,18);(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,26);(8,10)(20,24)(22,26);(3,4);(5,6)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22)(23,25)(24,26);(1,2)(3,4)(5,6);(11,12)(13,14)(15,17)(16,18)(19,21)(20,22)(23,25)(24,26);(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,25)(21,24)(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)(13,18)(14,16)(15,17), (11,13,15,16)(12,14,17,18), (11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,26), (8,10)(20,24)(22,26), (3,4), (5,6)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22)(23,25)(24,26), (1,2)(3,4)(5,6), (11,12)(13,14)(15,17)(16,18)(19,21)(20,22)(23,25)(24,26), (7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,25)(21,24)(23,26));