Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#43175 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C23×#392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
82805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(7,11)(9,12)(13,14)(15,19)(16,17)(18,20)(21,22,24,26)(23,27,28,25);(7,11)(9,12)(14,16)(18,19);(23,28)(25,27);(3,4);(5,6,8,10)(7,9,11,12)(13,15,17,20)(14,18,16,19)(21,23,24,28)(22,25,26,27);(1,2)(3,4);(5,7,8,11)(6,9,10,12)(13,16,17,14)(15,18,20,19)(21,22)(23,25)(24,26)(27,28)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((7,11)(9,12)(13,14)(15,19)(16,17)(18,20)(21,22,24,26)(23,27,28,25), (7,11)(9,12)(14,16)(18,19), (23,28)(25,27), (3,4), (5,6,8,10)(7,9,11,12)(13,15,17,20)(14,18,16,19)(21,23,24,28)(22,25,26,27), (1,2)(3,4), (5,7,8,11)(6,9,10,12)(13,16,17,14)(15,18,20,19)(21,22)(23,25)(24,26)(27,28));