Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C216×C42 |
| direct product | C22×#52410 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
86449 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(7,11)(10,12)(14,16)(15,19)(21,22,25,26)(23,28,24,27);(5,6)(7,12)(8,9)(10,11)(13,14)(15,18)(16,17)(19,20)(22,26)(27,28);(6,9)(7,11)(14,16)(18,20)(21,23,25,24)(22,27,26,28);(3,4);(5,6)(7,12)(8,9)(10,11)(13,15,17,19)(14,18,16,20)(21,22)(23,27)(24,28)(25,26);(1,2)(3,4);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,20,19,18)(21,24,25,23)(22,27,26,28)184 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((7,11)(10,12)(14,16)(15,19)(21,22,25,26)(23,28,24,27), (5,6)(7,12)(8,9)(10,11)(13,14)(15,18)(16,17)(19,20)(22,26)(27,28), (6,9)(7,11)(14,16)(18,20)(21,23,25,24)(22,27,26,28), (3,4), (5,6)(7,12)(8,9)(10,11)(13,15,17,19)(14,18,16,20)(21,22)(23,27)(24,28)(25,26), (1,2)(3,4), (5,7,8,11)(6,10,9,12)(13,16,17,14)(15,20,19,18)(21,24,25,23)(22,27,26,28));