Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C217×C4 |
| direct product | C22×#42502 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C22×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
53725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,8,9)(7,10,11,12)(21,22)(23,24)(25,27)(26,28);(7,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,23,26,25)(22,27,28,24);(6,9)(7,11)(13,15)(14,17)(16,20)(18,19)(21,22)(23,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,20)(17,19)(21,24,26,27)(22,23,28,25);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,19)(18,20)(21,25,26,23)(22,27,28,24);(3,4);(1,2)(3,4)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((5,6,8,9)(7,10,11,12)(21,22)(23,24)(25,27)(26,28), (7,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,23,26,25)(22,27,28,24), (6,9)(7,11)(13,15)(14,17)(16,20)(18,19)(21,22)(23,24)(25,27)(26,28), (5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,20)(17,19)(21,24,26,27)(22,23,28,25), (5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,19)(18,20)(21,25,26,23)(22,27,28,24), (3,4), (1,2)(3,4));