Finite Group DatabasebetaFinite Group Databasebeta| order | 328 |
| isoclinism | isoclinic class of D41 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C41 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D41 |
| commutator, hypocenter | C41 | C8 |
| Fitting, cosocle | C4×C41 | C2 |
| socle, prime radical | C2×C41 | C4 |
| LCS | G ▷ C41 | |
| UCS | C1 ◁ C4 | |
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)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43,44,46,45,47,48,49);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)Polycyclic presentation:
2^3,41|a^2=b,b^2=c,d^a=d^4088 conjugacy classes — view the full character table.
| maximal subgroup of | #10620270, C8×D41, #10620272, … |
| maximal normal subgroup of | #10620270, C8×D41, #10620272, … |
| maximal quotient of | #10620268, C2×#66491, C3×#66491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(328, 1);