Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#38246 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C22×#2410 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
69461 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)(7,10)(8,9)(11,12)(23,27)(25,28);(6,9)(10,11)(13,14,17,16)(15,19,20,18);(15,20)(18,19)(21,22,24,26)(23,25,27,28);(3,4);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,28)(24,26)(25,27);(1,2)(3,4);(5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,19,20,18)(21,23,24,27)(22,25,26,28)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)(7,10)(8,9)(11,12)(23,27)(25,28), (6,9)(10,11)(13,14,17,16)(15,19,20,18), (15,20)(18,19)(21,22,24,26)(23,25,27,28), (3,4), (5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,28)(24,26)(25,27), (1,2)(3,4), (5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,19,20,18)(21,23,24,27)(22,25,26,28));