Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
488 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 488 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:
(1,2,5,8,6,9,4,13)(3,10,11,15,12,7,14,16)(17,18)(19,27,29,31)(20,25)(21,24)(22,26)(23,28,32,30);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,27,32,31)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=gh,e^2=h,f^2=g,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #106206, #107483, #107554, … |
| maximal normal subgroup of | #106206, #107483, #107554, … |
| maximal quotient of | #160584, #160611, #160612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10666);