Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#43415 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C22×#2433 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
71997 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)(7,8)(9,10)(11,12)(15,19)(18,20)(22,25)(26,27);(6,10)(7,11)(13,14,17,16)(15,18,19,20)(21,22,24,25)(23,26,28,27);(15,19)(18,20)(21,22,24,25)(23,27,28,26);(3,4);(5,7,9,11)(6,8,10,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4);(5,8)(6,7)(9,12)(10,11)(13,16,17,14)(15,20,19,18)(21,23,24,28)(22,26,25,27)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)(7,8)(9,10)(11,12)(15,19)(18,20)(22,25)(26,27), (6,10)(7,11)(13,14,17,16)(15,18,19,20)(21,22,24,25)(23,26,28,27), (15,19)(18,20)(21,22,24,25)(23,27,28,26), (3,4), (5,7,9,11)(6,8,10,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,25)(27,28), (1,2)(3,4), (5,8)(6,7)(9,12)(10,11)(13,16,17,14)(15,20,19,18)(21,23,24,28)(22,26,25,27));