Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of #87771 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C52 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6038 |
| commutator, Fitting, cosocle | C4×C52 | C22 |
| Frattini, hypercenter | C4 | D52 |
| socle | C2×C52 | D4 |
| prime radical | C4⋊(C5⋊D5) | C2 |
| hypocenter | C52 | SD16 |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,3,4,2,5,6,8,7)(10,11)(12,13)(14,15,17,18,16);(1,2,5,7)(3,8,6,4)(9,10,12,13,11)(15,16)(17,18)Polycyclic presentation:
2^4,5^2|a^2=d,b^2=c,c^2=d,b^a=bc,c^a=cd,f^a=f^4,e^b=e^449 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#677, #11716912, #11716919, … |
| maximal normal subgroup of | D5×#677, #11716912, #11716919, … |
| maximal quotient of | #11716605, #11716607, #11716614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 68);