Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#49838 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
73217 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)(13,14,18,20)(15,19,16,17)(22,25)(23,27);(13,15)(14,19)(16,18)(17,20)(21,22)(23,28)(24,25)(26,27);(6,10)(8,12)(14,20)(15,16)(22,25)(26,28);(3,4);(5,7,9,11)(6,8,10,12)(13,16,18,15)(14,17,20,19)(21,23,24,27)(22,26,25,28);(1,2)(3,4);(5,8)(6,11)(7,10)(9,12)(13,17)(14,16)(15,20)(18,19)(21,22)(23,28)(24,25)(26,27)208 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)(13,14,18,20)(15,19,16,17)(22,25)(23,27), (13,15)(14,19)(16,18)(17,20)(21,22)(23,28)(24,25)(26,27), (6,10)(8,12)(14,20)(15,16)(22,25)(26,28), (3,4), (5,7,9,11)(6,8,10,12)(13,16,18,15)(14,17,20,19)(21,23,24,27)(22,26,25,28), (1,2)(3,4), (5,8)(6,11)(7,10)(9,12)(13,17)(14,16)(15,20)(18,19)(21,22)(23,28)(24,25)(26,27));