Finite Group DatabasebetaFinite Group Databasebeta| order | 625 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C53 · Schur multiplier (H2) = C54 |
| direct product | C5×H(3,𝔽5) |
| unambiguous semidirect product | C52⋊C52 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52 | C52 |
| commutator, Frattini, cosocle | C5 | C53 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52 ◁ G | |
| LECS | G ▷ C5 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 125 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3,4,5);(6,7,9,13,20)(8,11,15,22,27)(10,16,25,30,26)(12,18,24,21,28)(14,23,29,19,17);(7,10,17,18,11)(9,14,15,16,24)(13,21,25,22,23)(20,27,28,29,30)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(625, 12);