Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
393 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 75 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 — 3 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15,16,13)(12,14)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(2,5)(6,8)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,c^2=d,f^2=h,b^a=bd,c^a=ce,e^a=eg,c^b=ch,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #93708, #97216, #97217, … |
| maximal normal subgroup of | #93708, #97216, #97217, … |
| maximal quotient of | #99110, #99111, #99134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1690);