Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of #92539 |
| automorphism | Aut(G) = AGL(1,𝔽5)2⋊C2 (order 800) · Inn(G) = #1044 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽5) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C5 | #1044 |
| commutator, Fitting, hypocenter | H(3,𝔽5) | C4 |
| prime radical, cosocle | #6953 | C2 |
| LCS | G ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C5 | |
29 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 25 |
A minimal faithful permutation representation — 2 generators on 25 points:
(2,4,5,6)(7,15,16,17)(8,11,12,14)(10,21,22,19)(13,18,23,25);(1,2,17,10,16)(3,8,6,23,5)(4,21,20,18,22)(7,13,11,25,24)(9,19,14,15,12)Polycyclic presentation:
2^2,5^3|a^2=b,c^a=c^3,d^a=d^2,c^b=c^4,d^b=d^4,d^c=de26 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 25);