Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#45362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C23×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
58405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6)(7,8)(9,11)(10,12)(13,14,18,17)(15,19,16,20);(5,7)(6,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,18)(22,25)(23,27);(7,9)(8,11)(13,15,18,16)(14,19,17,20)(21,22)(23,26)(24,25)(27,28);(5,8,10,11)(6,7,12,9)(13,16,18,15)(14,19,17,20)(21,23,24,27)(22,26,25,28);(5,9,10,7)(6,11,12,8)(13,17,18,14)(15,19,16,20);(3,4);(1,2)(3,4)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)(9,11)(10,12)(13,14,18,17)(15,19,16,20), (5,7)(6,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,18)(22,25)(23,27), (7,9)(8,11)(13,15,18,16)(14,19,17,20)(21,22)(23,26)(24,25)(27,28), (5,8,10,11)(6,7,12,9)(13,16,18,15)(14,19,17,20)(21,23,24,27)(22,26,25,28), (5,9,10,7)(6,11,12,8)(13,17,18,14)(15,19,16,20), (3,4), (1,2)(3,4));