Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#55731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
75557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,9,10)(7,8,11,12)(13,14)(15,17)(16,20)(18,19)(21,22,24,27)(23,26,28,25);(13,15)(14,17)(16,18)(19,20)(21,23)(22,25)(24,28)(26,27);(7,11)(8,12)(15,16)(17,20)(21,23)(22,26)(24,28)(25,27);(3,4);(5,7,9,11)(6,8,10,12)(13,16,18,15)(14,17,19,20)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4);(5,8)(6,7)(9,12)(10,11)(13,17)(14,15)(16,19)(18,20)(21,22)(23,25)(24,27)(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((5,6,9,10)(7,8,11,12)(13,14)(15,17)(16,20)(18,19)(21,22,24,27)(23,26,28,25), (13,15)(14,17)(16,18)(19,20)(21,23)(22,25)(24,28)(26,27), (7,11)(8,12)(15,16)(17,20)(21,23)(22,26)(24,28)(25,27), (3,4), (5,7,9,11)(6,8,10,12)(13,16,18,15)(14,17,19,20)(21,23)(22,26)(24,28)(25,27), (1,2)(3,4), (5,8)(6,7)(9,12)(10,11)(13,17)(14,15)(16,19)(18,20)(21,22)(23,25)(24,27)(26,28));