Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of #92534 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | C5×C25 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C5×C25 | C4 |
| Frattini | C5 | #1042 |
| socle | C52 | AGL(1,𝔽5) |
| prime radical, cosocle | C5×D25 | C2 |
| LCS | G ▷ C5×C25 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 2 generators on 30 points:
(2,3)(4,5)(7,9,10,16)(8,13,14,15)(11,20,21,19)(12,24,25,30)(17,23,26,18)(22,29,28,27);(1,2,4,5,3)(6,12,16,21,19,15,27,30,22,17,14,7,23,18,10,8,26,28,24,29,13,20,11,9,25)Polycyclic presentation:
2^2,5^3|a^2=b,d^5=e^2,c^a=c^4,d^a=d^3e,e^a=e^3,d^b=d^4e^3,e^b=e^438 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1035, C2×#92534, C3×#92534, … |
| maximal normal subgroup of | D5×#1035, C2×#92534, C3×#92534, … |
| maximal quotient of | #11758644, C2×#92534, C3×#92534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 20);