Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×M32 ⋊ C2 |
| Sylow | C5 ⋊ #509 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D40 |
| commutator, cosocle | C5×C8 | C23 |
| Frattini | C8 | C22×D5 |
| Fitting | C5×M32 | C2 |
| socle | C2×C5 | C2×D8 |
| hypercenter | M32 | D5 |
| hypocenter | C5 | #509 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(6,7,13,19,9,14,16,17,10,12,20,21,15,18,8,11);(7,12)(11,17)(14,18)(19,21);(2,3)(4,5)(7,11)(8,13)(9,15)(12,17)(14,21)(16,20)(18,19)Polycyclic presentation:
2^6,5|c^2=de,d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^4,c^b=cf56 conjugacy classes — view the full character table.
| maximal subgroup of | #10599884, #10599913, #10599919, … |
| maximal normal subgroup of | #10599884, #10599913, #10599919, … |
| maximal quotient of | #10599862, #10599870, #10599871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 535);