Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
486 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,27)(20,25)(21,24)(22,26)(23,28)(29,31)(30,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,31)(26,32);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,32,31);(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,31,32,27)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=h,e^2=h,f^2=gh,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #106020, #107607, #107626, … |
| maximal normal subgroup of | #106020, #107607, #107626, … |
| maximal quotient of | #160536, #160547, #160555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10728);