Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of #87770 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C52 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6037 |
| commutator, Fitting, cosocle | C4×C52 | C22 |
| Frattini, hypercenter | C4 | D52 |
| socle | C2×C52 | D4 |
| hypocenter | C52 | D8 |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,5)(3,8)(6,7)(10,11)(12,13)(14,15,17,18,16);(1,2)(3,5)(4,7)(6,8)(9,10,12,13,11)(15,16)(17,18)Polycyclic presentation:
2^4,5^2|c^2=d,b^a=bc,c^a=cd,f^a=f^4,c^b=cd,e^b=e^443 conjugacy classes — view the full character table.
| maximal subgroup of | #11716792, #11716793, #11716915, … |
| maximal normal subgroup of | #11716792, #11716793, #11716915, … |
| maximal quotient of | #11716586, #11716588, #11716591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 64);