Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of #92537 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽5) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | H(3,𝔽5) | C4 |
| Frattini, socle | C5 | #1043 |
| prime radical, cosocle | #6953 | C2 |
| LCS | G ▷ H(3,𝔽5) | |
| UCS | C1 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 25 |
A minimal faithful permutation representation — 3 generators on 25 points:
(2,4,5,10)(3,8)(6,12,14,20)(7,15,13,24)(9,18)(11,21,22,17)(16,23,25,19);(2,6,16,17,7)(4,11,12,15,23)(5,13,21,25,14)(10,19,24,20,22);(1,2,4,10,5)(3,7,12,20,13)(6,15,24,14,8)(9,17,23,19,21)(11,22,25,18,16)Polycyclic presentation:
2^2,5^3|a^2=b,c^a=c^2,d^a=d^2,e^a=e^4,c^b=c^4,d^b=d^4,d^c=de14 conjugacy classes — view the full character table.
| maximal subgroup of | #11758701, #92537⋊C2, C2×#92537, … |
| maximal normal subgroup of | #11758701, #92537⋊C2, C2×#92537, … |
| maximal quotient of | #11758647, C2×#92537, C3×#92537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 23);