Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of C5⋊D25 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
No aliases for this group.
| derived, Sylow | C5×C25 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C5⋊D25 |
| commutator, hypocenter | C5×C25 | C4 |
| Frattini | C2×C5 | C5⋊D5 |
| Fitting, cosocle | C2×C5×C25 | C2 |
| socle, prime radical | C2×C52 | D5 |
| LCS | G ▷ C5×C25 | |
| UCS | C1 ◁ C2 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 500 |
A minimal faithful permutation representation — 3 generators on 34 points:
(2,3)(4,5)(7,9)(8,12)(10,15)(11,18)(13,21)(14,20)(16,17)(19,25)(22,26)(23,27)(24,29)(28,30)(31,32,33,34);(1,2,4,5,3);(6,20,30,23,9,12,26,24,15,18,21,19,17,16,25,13,11,10,29,22,8,7,27,28,14)Polycyclic presentation:
2^2,5^3|a^2=b,c^5=e^4,c^a=c^4e,d^a=d^4,e^a=e^4128 conjugacy classes — view the full character table.
| maximal subgroup of | #11758646, #11758648, #11758650, … |
| maximal normal subgroup of | #11758646, #11758648, #11758650, … |
| maximal quotient of | #11758634, C2×#92524, C3×#92524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 10);