Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9159 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #386 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,11,12,7,14,15,13,16)(17,18,23,26,21,20,22,25)(19,27,29,31,30,24,28,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19,22,29,21,30,23,28)(18,24,26,31,20,27,25,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,21,18)(19,27,30,24)(22,25,23,26)(28,31,29,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eg,f^c=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9168, #197124, #197128, … |
| maximal normal subgroup of | C2×#9168, #197124, #197128, … |
| maximal quotient of | #99789, #99790, #99803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2156);