Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C225 |
| direct product | C22×#9164052 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C26 |
| commutator, Frattini, cosocle | C23 | C28 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
943132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 8 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(14,17)(18,20)(23,28)(24,26);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,27)(26,28);(14,17)(15,19)(23,28)(24,26);(5,6,8,10)(7,11,12,9)(13,15)(14,18)(16,19)(17,20)(21,23,25,28)(22,26,27,24);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,26)(27,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,28)(26,27);(3,4);(1,2)(3,4)356 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((5,6)(7,11)(8,10)(9,12)(14,17)(18,20)(23,28)(24,26), (5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,27)(26,28), (14,17)(15,19)(23,28)(24,26), (5,6,8,10)(7,11,12,9)(13,15)(14,18)(16,19)(17,20)(21,23,25,28)(22,26,27,24), (5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,26)(27,28), (5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,28)(26,27), (3,4), (1,2)(3,4));