Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×C16×AGL(1,𝔽5) (order 640) · Inn(G) = D5 · Out(G) = C22×C16 |
| homology | Abelianization (H1) = C64 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C5 ⋊ C64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | D5 |
| commutator, hypocenter | C5 | C64 |
| Fitting, cosocle | C5×C32 | C2 |
| socle, prime radical | C2×C5 | C32 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C32 | |
14 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(2,3)(4,5)(6,7,8,13,9,14,18,28,10,15,19,29,22,32,38,48,11,16,20,30,23,33,39,49,25,35,41,51,44,54,58,63,12,17,21,31,24,34,40,50,26,36,42,52,45,55,59,64,27,37,43,53,46,56,60,65,47,57,61,66,62,67,68,69);(1,2,4,5,3)Polycyclic presentation:
2^6,5|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,g^a=g^4128 conjugacy classes — view the full character table.
| maximal subgroup of | #10597921, D5×C64, #10597923, … |
| maximal normal subgroup of | #10597921, D5×C64, #10597923, … |
| maximal quotient of | #10597919, C2×#64663, C3×#64663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1);