Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C53 · Schur multiplier (H2) = C54 |
| direct product | C2×C5×H(3,𝔽5) |
| unambiguous semidirect product | (C2×C52)⋊C52 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C52 | C52 |
| commutator, Frattini, cosocle | C5 | C2×C53 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C2×C52 ◁ G | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 250 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,4,5,6,7);(8,9,11,15,22)(10,13,17,24,29)(12,18,27,32,28)(14,20,26,23,30)(16,25,31,21,19);(9,12,19,20,13)(11,16,17,18,26)(15,23,27,24,25)(22,29,30,31,32)290 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 12);