Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | Aut(G) = C23×#45355 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3207 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,21,24)(19,22)(20,23,28,25)(26,31)(27,29)(30,32);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19)(18,22,24,31)(20,27)(21,26)(23,29,25,30)(28,32);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9281, #198784, #198786, … |
| maximal normal subgroup of | C2×#9281, #198784, #198786, … |
| maximal quotient of | #99935, #99938, #99943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2269);