Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C29 · Schur multiplier (H2) = C234 |
| direct product | C22×#10586564 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C27 |
| commutator, Frattini, cosocle | C22 | C29 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
9165360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 9 generators on 28 points:
(5,6,9,12)(7,8,11,10)(13,14)(15,20)(16,17)(18,21)(19,25)(22,24)(23,27)(26,28);(7,11)(8,10)(13,15)(14,19)(16,23)(17,24)(18,25)(20,21)(22,26)(27,28);(5,7)(6,10)(8,12)(9,11);(5,8,9,10)(6,11,12,7)(13,16,18,26)(14,17,21,28)(15,22,25,23)(19,27,20,24);(5,6)(7,8)(9,12)(10,11)(13,16)(14,17)(15,23)(18,26)(19,24)(20,27)(21,28)(22,25);(5,6)(7,8)(9,12)(10,11)(13,15)(14,20)(16,23)(17,27)(18,25)(19,21)(22,26)(24,28);(5,6)(7,8)(9,12)(10,11)(13,17)(14,16)(15,24)(18,28)(19,23)(20,22)(21,26)(25,27);(3,4);(1,2)(3,4)560 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((5,6,9,12)(7,8,11,10)(13,14)(15,20)(16,17)(18,21)(19,25)(22,24)(23,27)(26,28), (7,11)(8,10)(13,15)(14,19)(16,23)(17,24)(18,25)(20,21)(22,26)(27,28), (5,7)(6,10)(8,12)(9,11), (5,8,9,10)(6,11,12,7)(13,16,18,26)(14,17,21,28)(15,22,25,23)(19,27,20,24), (5,6)(7,8)(9,12)(10,11)(13,16)(14,17)(15,23)(18,26)(19,24)(20,27)(21,28)(22,25), (5,6)(7,8)(9,12)(10,11)(13,15)(14,20)(16,23)(17,27)(18,25)(19,21)(22,26)(24,28), (5,6)(7,8)(9,12)(10,11)(13,17)(14,16)(15,24)(18,28)(19,23)(20,22)(21,26)(25,27), (3,4), (1,2)(3,4));