Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of #92566 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C53 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C53 | C22 |
| LCS | G ▷ C53 | |
| UCS | C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(3,4)(7,8)(9,10)(11,12,14,15,13);(2,3)(4,5)(6,7,9,10,8)(12,13)(14,15)Polycyclic presentation:
2^2,5^3|c^a=c^4,e^a=e^4,c^b=c^4,d^b=d^444 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 52);