Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of D52⋊C2 |
| automorphism | Aut(G) = C22×(D5⋊AGL(1,𝔽5))⋊C2 (order 1600) · Inn(G) = D52⋊C2 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D52⋊C2 |
| commutator | C5⋊D5 | C2×C4 |
| Fitting, socle | C2×C52 | D4 |
| prime radical | C2×D52 | C2 |
| cosocle | C2×C5⋊D5 | C22 |
| hypocenter | C52 | C22⋊C4 |
| LCS | G ▷ C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C2 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,9,3,7,8)(2,5)(4,10)(11,12,13,14);(1,2)(3,6)(4,9)(5,8)(7,10)Polycyclic presentation:
2^4,5^2|b^2=d,b^a=bc,e^a=f^3,f^a=e^2,e^b=f,f^b=e,e^c=e^4,f^c=f^428 conjugacy classes — view the full character table.
| maximal subgroup of | #11717306, #11717307, #11717310, … |
| maximal normal subgroup of | #11717306, #11717307, #11717310, … |
| maximal quotient of | #11716784, #11716785, #11716786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 129);