Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of #6053 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C52 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #6053 |
| commutator, hypocenter | C52 | C16 |
| Fitting, socle, prime radical | C2×C52 | C8 |
| cosocle | #6033 | C2 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C2 | |
22 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)(3,4,5,7,6,8,9,10)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(1,3,5,9,6)Polycyclic presentation:
2^4,5^2|a^2=b,b^2=c,c^2=d,e^a=e^2f^4,f^a=e^2f^3,e^b=e^2,f^b=f^2,e^c=e^4,f^c=f^422 conjugacy classes — view the full character table.
| maximal subgroup of | #11717315, #11717316, C2×#87822, … |
| maximal normal subgroup of | #11717315, #11717316, C2×#87822, … |
| maximal quotient of | #11716729, C2×#87822, C3×#87822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 116);