Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C217×C4 |
| direct product | C22×#38942 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
63513 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)(7,11)(8,9)(10,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,24)(23,28,27,25);(6,9)(10,11)(14,16)(15,19)(21,23)(22,25)(24,28)(26,27);(13,15)(14,17)(16,20)(18,19)(21,23)(22,25)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,20,19,17)(21,24,26,22)(23,28,27,25);(5,7,8,12)(6,10,9,11)(13,17,18,20)(14,19,16,15)(21,25,26,28)(22,27,24,23);(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)(7,11)(8,9)(10,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,24)(23,28,27,25), (6,9)(10,11)(14,16)(15,19)(21,23)(22,25)(24,28)(26,27), (13,15)(14,17)(16,20)(18,19)(21,23)(22,25)(24,28)(26,27), (5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,20,19,17)(21,24,26,22)(23,28,27,25), (5,7,8,12)(6,10,9,11)(13,17,18,20)(14,19,16,15)(21,25,26,28)(22,27,24,23), (3,4), (1,2)(3,4));