Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C227 |
| direct product | C2×#72268588 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C27 |
| commutator, Frattini, cosocle | C24 | C28 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
70754099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 8 generators on 26 points:
(4,6)(8,10)(11,12,14,17)(13,15,18,16)(19,20,22,23)(21,25,26,24);(11,13)(12,15)(14,18)(16,17)(20,23)(24,25);(8,10)(11,13)(12,16)(14,18)(15,17)(19,21)(20,24)(22,26)(23,25);(1,2);(8,10);(19,21)(20,25)(22,26)(23,24);(11,13)(12,15)(14,18)(16,17)(19,21)(20,25)(22,26)(23,24);(3,4)(5,6)(7,8,9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,24)(22,23)(25,26)640 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((4,6)(8,10)(11,12,14,17)(13,15,18,16)(19,20,22,23)(21,25,26,24), (11,13)(12,15)(14,18)(16,17)(20,23)(24,25), (8,10)(11,13)(12,16)(14,18)(15,17)(19,21)(20,24)(22,26)(23,25), (1,2), (8,10), (19,21)(20,25)(22,26)(23,24), (11,13)(12,15)(14,18)(16,17)(19,21)(20,25)(22,26)(23,24), (3,4)(5,6)(7,8,9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,24)(22,23)(25,26));