Finite Group DatabasebetaFinite Group Databasebeta| order | 25 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = GL(2,𝔽5) (order 480) · Inn(G) = C1 · Out(G) = GL(2,𝔽5) |
| holomorph | G ⋊ Aut(G) = AGL(2,𝔽5) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C5 |
| abelian | C52 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 25 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2,3,4,5);(6,7,8,9,10)25 conjugacy classes — view the full character table.
| center of | C52×D3, C52×D5, C52×A4, … |
| derived subgroup of | C5⋊D5, C52⋊C3, #1039, … |
| Frattini subgroup of | C252, C52⋊C25, C25⋊C25, … |
| Fitting subgroup of | C5×D5, C5⋊D5, C52⋊C3, … |
| socle of | C5×D5, C5⋊D5, C52⋊C3, … |
| maximal subgroup of | C5×D5, C5⋊D5, C2×C52, … |
| maximal normal subgroup of | C5×D5, C5⋊D5, C2×C52, … |
| maximal quotient of | C2×C52, C3×C52, C5×C25, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25, 2);