Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of #87837 |
| automorphism | Aut(G) = C2×#11717307 (order 1600) · Inn(G) = D52⋊C2 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C52 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D52⋊C2 |
| commutator, cosocle | #1039 | C22 |
| Fitting, socle | C2×C52 | D4 |
| prime radical | #6037 | C2 |
| hypocenter | C52 | SD16 |
| LCS | G ▷ #1039 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,9,3,7,8)(2,5)(4,10)(11,12,14,16)(13,18,17,15);(1,2)(3,6)(4,9)(5,8)(7,10)(12,15)(13,17)(16,18)Polycyclic presentation:
2^4,5^2|b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=f^3,f^a=e^2,c^b=cd,e^b=f,f^b=e,e^c=e^4,f^c=f^425 conjugacy classes — view the full character table.
| maximal subgroup of | #11717381, #11717382, #11717384, … |
| maximal normal subgroup of | #11717381, #11717382, #11717384, … |
| maximal quotient of | #11716797, #11716798, #11716800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 132);