Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C229 |
| direct product | C23×#846041 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C25 |
| commutator, Frattini, cosocle | C24 | C28 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
At least 7548 conjugacy classes of subgroups; only the 610683 normal subgroups are shown.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 26 points:
(12,13)(15,16)(17,18)(21,25)(23,26);(7,8)(9,10)(11,12)(13,14)(16,18)(19,20,22,24)(21,23,25,26);(3,4);(5,6);(8,10)(12,13)(15,16)(17,18)(21,25)(23,26);(1,2)(3,4)(5,6);(12,13)(19,20)(21,23)(22,24)(25,26);(7,8)(9,10)(11,13,14,12)(19,21)(20,23)(22,25)(24,26)736 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((12,13)(15,16)(17,18)(21,25)(23,26), (7,8)(9,10)(11,12)(13,14)(16,18)(19,20,22,24)(21,23,25,26), (3,4), (5,6), (8,10)(12,13)(15,16)(17,18)(21,25)(23,26), (1,2)(3,4)(5,6), (12,13)(19,20)(21,23)(22,24)(25,26), (7,8)(9,10)(11,13,14,12)(19,21)(20,23)(22,25)(24,26));