Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C217×C4 |
| direct product | C22×#52966 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
65665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 28 points:
(6,8)(7,11)(13,14,18,19)(15,16,17,20)(23,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,18)(19,20)(22,25)(26,28);(6,8)(7,11)(14,19)(15,17)(21,22)(23,28)(24,25)(26,27);(5,7,9,11)(6,10,8,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,26)(24,27)(25,28);(5,8,9,6)(7,12,11,10)(13,17,18,15)(14,16,19,20);(3,4);(1,2)(3,4)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((6,8)(7,11)(13,14,18,19)(15,16,17,20)(23,27)(26,28), (5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,18)(19,20)(22,25)(26,28), (6,8)(7,11)(14,19)(15,17)(21,22)(23,28)(24,25)(26,27), (5,7,9,11)(6,10,8,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,26)(24,27)(25,28), (5,8,9,6)(7,12,11,10)(13,17,18,15)(14,16,19,20), (3,4), (1,2)(3,4));