Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| 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 | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
217 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,7,10)(3,6,9,5)(4,11)(8,12)(13,15)(14,16)(17,18,21,26,22,25,20,24)(19,23,27,30,28,31,29,32);(1,3,5,2,7,9,6,10)(4,11,13,8,14,16,12,15)(17,19,20,27,22,28,21,29)(18,23,26,32,25,31,24,30);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20,22,21)(18,24,25,26)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=egh,f^c=fgh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9250, #198260, #198265, … |
| maximal normal subgroup of | C2×#9250, #198260, #198265, … |
| maximal quotient of | #99889, #99890, #99905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2238);