Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| direct product | C2×#72339320 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C27 |
| commutator, Frattini, cosocle | C24 | C28 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
17461987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 26 points:
(8,10)(19,20)(21,25)(22,23)(24,26);(11,12)(13,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26);(4,6)(8,10)(11,13,14,18)(12,15,17,16);(4,6)(13,18)(15,16);(1,2);(19,21)(20,25)(22,24)(23,26);(11,12)(13,15)(14,17)(16,18)(19,21)(20,25)(22,24)(23,26);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,22,23,20)(21,25,26,24)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)(19,20)(21,25)(22,23)(24,26), (11,12)(13,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26), (4,6)(8,10)(11,13,14,18)(12,15,17,16), (4,6)(13,18)(15,16), (1,2), (19,21)(20,25)(22,24)(23,26), (11,12)(13,15)(14,17)(16,18)(19,21)(20,25)(22,24)(23,26), (3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,22,23,20)(21,25,26,24));