Finite Group DatabasebetaFinite Group Databasebeta| order | 328 |
| isoclinism | isoclinic class of #4223 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C41 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #4223 |
| commutator, hypocenter | C41 | C8 |
| Fitting, socle, prime radical | C2×C41 | C4 |
| cosocle | Dic41 | C2 |
| LCS | G ▷ C41 | |
| UCS | C1 ◁ C2 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 328 |
A minimal faithful permutation representation — 2 generators on 49 points:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,35)(23,33,34,40)(27,37,38,36)(29,41,39,32)(42,43,44,46,45,47,48,49);(1,2,5,10,19,30,34,22,13,6,11,20,32,33,17,29,38,26,15,24,36,37,21,8,16,27,39,28,40,41,25,12,3,7,14,23,35,31,18,9,4)Polycyclic presentation:
2^3,41|a^2=b,b^2=c,d^a=d^32,d^b=d^4028 conjugacy classes — view the full character table.
| maximal subgroup of | #10620295, #10620296, #10620297, … |
| maximal normal subgroup of | #10620295, #10620296, #10620297, … |
| maximal quotient of | #10620270, C2×#66493, C3×#66493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(328, 3);