Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of #1042 |
| automorphism | Aut(G) = C4×AGL(1,𝔽5)2 (order 1600) · Inn(G) = #1042 · Out(G) = C42 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C52 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #1042 |
| commutator, hypocenter | C52 | C16 |
| Fitting | C4×C52 | C4 |
| socle, prime radical | C2×C52 | C8 |
| cosocle | C5×#168 | C2 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C4 | |
23 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,19)(20,21)(23,24,25,26);(17,18,20,21,19)(22,23,26,24,25)Polycyclic presentation:
2^4,5^2|a^2=b,b^2=c,c^2=d,e^a=e^4,f^a=f^3,f^b=f^452 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#674, #11716747, #11716748, … |
| maximal normal subgroup of | D5×#674, #11716747, #11716748, … |
| maximal quotient of | #11716579, C2×#87763, C3×#87763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 57);