Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#40720 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
79725 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)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,25)(26,28);(13,15,16,20)(14,17,18,19)(21,22,24,25)(23,26,28,27);(6,7)(10,11)(14,18)(15,20)(22,25)(26,27);(3,4);(5,7,9,6)(8,10,12,11)(13,15)(14,19)(16,20)(17,18)(21,23,24,28)(22,26,25,27);(1,2)(3,4);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,26)(24,25)(27,28)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)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,25)(26,28), (13,15,16,20)(14,17,18,19)(21,22,24,25)(23,26,28,27), (6,7)(10,11)(14,18)(15,20)(22,25)(26,27), (3,4), (5,7,9,6)(8,10,12,11)(13,15)(14,19)(16,20)(17,18)(21,23,24,28)(22,26,25,27), (1,2)(3,4), (5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,26)(24,25)(27,28));