Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D52 |
| commutator, socle, cosocle | C2×C52 | C23 |
| Fitting | C4×C52 | C22 |
| hypercenter | C4 | D52 |
| hypocenter | C52 | #41 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,3,4,7)(2,5,6,8)(9,10,12,13,11)(14,15,17,18,16);(1,2,4,6)(3,8,7,5)(10,11)(12,13);(3,7)(5,8)(15,16)(17,18)Polycyclic presentation:
2^4,5^2|b^2=d,c^2=d,c^a=cd,f^a=f^4,c^b=cd,e^b=e^446 conjugacy classes — view the full character table.
| maximal subgroup of | #11716769, #11716770, #11716916, … |
| maximal normal subgroup of | #11716769, #11716770, #11716916, … |
| maximal quotient of | #11716961, #11716964, #11716966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 165);