Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of #92536 |
| 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 | #1043 |
| socle | C52 | AGL(1,𝔽5) |
| prime radical, cosocle | C5⋊D25 | C2 |
| LCS | G ▷ C5×C25 | |
| UCS | C1 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 125 |
A minimal faithful permutation representation — 3 generators on 30 points:
(2,3,4,5)(7,9,10,15)(8,13,14,25)(11,19,20,24)(12,22,23,16)(17,27,28,30)(18,21,26,29);(1,2,3,5,4);(6,7,11,21,18,8,12,9,16,27,13,24,17,28,19,25,30,22,15,23,14,26,29,20,10)Polycyclic presentation:
2^2,5^3|a^2=b,c^5=e,c^a=c^2e,d^a=d^2,e^a=e^2,c^b=c^4e^4,d^b=d^4,e^b=e^435 conjugacy classes — view the full character table.
| maximal subgroup of | D5⋊#1035, C2×#92536, C3×#92536, … |
| maximal normal subgroup of | D5⋊#1035, C2×#92536, C3×#92536, … |
| maximal quotient of | #11758646, C2×#92536, C3×#92536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 22);