Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | Aut(G) = C2×#66297 (order 640) · Inn(G) = (C24⋊C5)⋊C2 · Out(G) = C22 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | (C24⋊C5)⋊C2 |
| commutator, prime radical, cosocle, hypocenter | 2−1+4⋊C5 | C2 |
| Fitting | 2−1+4 | D5 |
| LCS | G ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,23,58,10)(2,36,64,27)(3,11,56,43)(4,41,55,39)(5,12,45,52)(6,54,44,32)(7,53,46,40)(8,30,62,47)(9,16,61,19)(13,50,63,25)(14,35,57,28)(15,34,21,37)(17,42,51,59)(18,29,60,26)(20,49,24,38)(22,31,48,33);(3,14,25,58,30)(4,19,50,44,16)(5,24,55,56,36)(6,17,20,15,46)(8,13,38,64,42)(9,33,49,43,31)(10,37,61,62,53)(11,32,34,28,52)(18,39,35,59,41)(21,47,29,22,26)(23,40,63,60,54)(27,57,48,51,45)Polycyclic presentation:
2,5,2^5|a^2=g,d^2=g,e^2=g,b^a=b^4,c^a=cd,d^a=dg,e^a=cdf,f^a=ce,c^b=cd,d^b=de,e^b=ef,f^b=cg,d^c=dg,e^c=eg,f^c=fg,e^d=eg14 conjugacy classes — view the full character table.
| maximal subgroup of | #10619372, C2×#66243, #10619380, … |
| maximal normal subgroup of | #10619372, C2×#66243, #10619380, … |
| maximal quotient of | #10617013, #10617016, C2×#66243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1581);