Finite Group DatabasebetaFinite Group Databasebeta| order | 1250 |
| isoclinism | isoclinic class of #11929378 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽5)⋊C5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽5)⋊C5 | C2 |
| Frattini | C52 | C5⋊D5 |
| socle | C5 | #6953 |
| LCS | G ▷ H(3,𝔽5)⋊C5 | |
| UCS | C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 25 |
Polycyclic presentation:
2,5^4|b^a=b^4,c^a=c^4,d^a=de^4,e^a=e^4,c^b=cd,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #72209620 |
| maximal quotient of | #72209680 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1250, 29);