Finite Group DatabasebetaFinite Group Databasebeta| order | 625 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 6000000 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C53 |
| abelian | C52×C25 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C5 | C53 |
| socle, prime radical | C53 | C5 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C5 ▷ C1 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 625 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25);(26,27,28,29,30);(31,32,33,34,35)625 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(625, 11);