Finite Group DatabasebetaFinite Group Databasebeta| order | 328 |
| isoclinism | isoclinic class of #66502 |
| automorphism | Aut(G) = AGL(1,𝔽41) (order 1640) · Inn(G) = G · Out(G) = C5 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C41 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C41 | C8 |
| prime radical | D41 | C4 |
| cosocle | #4223 | C2 |
| LCS | G ▷ C41 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 41 |
A minimal faithful permutation representation — 2 generators on 41 points:
(2,3,4,7,5,8,10,15)(6,12,13,26,14,9,18,19)(11,22,23,36,24,35,21,27)(16,28,29,34,30,17,31,20)(25,32,37,40,38,39,33,41);(1,2,6,7,16,29,26,38,24,10,21,35,34,12,8,17,32,36,18,33,40,41,37,13,27,39,28,3,9,20,22,23,4,11,25,19,31,30,15,14,5)Polycyclic presentation:
2^3,41|a^2=b,b^2=c,d^a=d^27,d^b=d^32,d^c=d^4013 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#66502, C3×#66502, C3⋊#66502, … |
| maximal normal subgroup of | C2×#66502, C3×#66502, AGL(1,𝔽41), … |
| maximal quotient of | #10620295, C2×#66502, C3×#66502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(328, 12);