Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| isoclinism | isoclinic class of #6118 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C13 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×#261 |
| commutator | C2×C13 | C42 |
| Fitting | C2×C4×C13 | C4 |
| socle, prime radical | C22×C13 | C2×C4 |
| cosocle | C2×Dic13 | C22 |
| hypercenter | C2×C4 | #261 |
| hypocenter | C13 | #97 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,5,8,12,6,10,11,3,7,13,9,4)(14,15,16,17)(19,22)(23,25);(2,3,4,6)(5,8,9,13)(7,11,12,10)(18,19,20,23,21,22,24,25)Polycyclic presentation:
2^5,13|a^2=c,b^2=d,c^2=e,b^a=be,f^a=f^8,f^c=f^1244 conjugacy classes — view the full character table.
| maximal subgroup of | #11718242, #11718293, #11718299, … |
| maximal normal subgroup of | #11718242, #11718293, #11718299, … |
| maximal quotient of | #11718232, #1057⋊C8, #11718262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 77);