Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of C5⋊(C5⋊D5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C53 |
No aliases for this group.
| derived, Sylow | C53 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C5⋊(C5⋊D5) |
| commutator, hypocenter | C53 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C53 | C2 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C2 | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 500 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3,4)(6,7)(8,9)(11,12)(13,14)(16,17)(18,19);(15,16,18,19,17);(10,11,13,14,12);(5,6,8,9,7)Polycyclic presentation:
2^2,5^3|a^2=b,c^a=c^4,d^a=d^4,e^a=e^4128 conjugacy classes — view the full character table.
| maximal subgroup of | #11758742, #11758743, D5×#1039, … |
| maximal normal subgroup of | #11758742, #11758743, D5×#1039, … |
| maximal quotient of | #11758733, C2×#92553, C3×#92553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 39);