Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C217×C4 |
| direct product | C22×#39933 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
68253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,17,18)(15,19,20,16)(21,22,25,28)(23,24,27,26);(14,18)(16,19)(23,27)(24,26);(6,9)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,26)(24,28)(25,27);(5,7,8,12)(6,10,9,11)(13,14)(15,16)(17,18)(19,20)(21,24,25,26)(22,27,28,23);(5,6)(7,11)(8,9)(10,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,28)(26,27);(3,4);(1,2)(3,4)232 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((5,6,8,9)(7,11,12,10)(13,14,17,18)(15,19,20,16)(21,22,25,28)(23,24,27,26), (14,18)(16,19)(23,27)(24,26), (6,9)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,26)(24,28)(25,27), (5,7,8,12)(6,10,9,11)(13,14)(15,16)(17,18)(19,20)(21,24,25,26)(22,27,28,23), (5,6)(7,11)(8,9)(10,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,28)(26,27), (3,4), (1,2)(3,4));