Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×H(5,𝔽2) ⋊ C2 |
| Sylow | C5 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4099 |
| commutator, cosocle | C23×C5 | C23 |
| Frattini | C23 | C22×D5 |
| Fitting | C5×H(5,𝔽2) | C2 |
| socle | C2×C5 | #120 |
| hypercenter | H(5,𝔽2) | D5 |
| hypocenter | C5 | B(4,𝔽2) |
| LCS | G ▷ C23×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ H(5,𝔽2) | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,4,5,3)(6,8)(7,10)(9,12)(11,13);(6,7)(8,10)(9,11)(12,13);(2,3)(4,5)(10,13)Polycyclic presentation:
2^6,5|b^a=be,c^a=cd,g^a=g^4,d^b=df,e^c=ef50 conjugacy classes — view the full character table.
| maximal subgroup of | #10601777, #10601778, #10601783, … |
| maximal normal subgroup of | #10601777, #10601778, #10601783, … |
| maximal quotient of | #10602447, #10602450, #10602638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 871);