Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C2×C52 |
| direct product | C2×C4×H(3,𝔽5) |
| unambiguous semidirect product | (C2×C4×C52)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C52 |
| commutator | C5 | C2×C4×C52 |
| Frattini, cosocle | C2×C5 | C22×C52 |
| socle | C22×C5 | C2×C52 |
| prime radical | C22×H(3,𝔽5) | C2 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2)(7,8,10,14,21)(9,12,16,23,28)(11,17,26,31,27)(13,19,25,22,29)(15,24,30,20,18);(3,4,5,6)(8,11,18,19,12)(10,15,16,17,25)(14,22,26,23,24)(21,28,29,30,31)232 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 78);