Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of #92559 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C53 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C53 | C4 |
| prime radical, cosocle | C52×D5 | C2 |
| LCS | G ▷ C53 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 3 generators on 15 points:
(2,3)(4,5)(7,8)(9,10)(12,13,14,15);(6,7,9,10,8)(11,12,15,13,14);(1,2,4,5,3)Polycyclic presentation:
2^2,5^3|a^2=b,c^a=c^4,d^a=d^4,e^a=e^3,e^b=e^453 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×C5⋊D5, D5×#1042, C2×#92559, … |
| maximal normal subgroup of | AGL(1,𝔽5)×C5⋊D5, D5×#1042, C2×#92559, … |
| maximal quotient of | #11758739, C2×#92559, C3×#92559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 45);