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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
198 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 198 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,8,11)(3,9,10,12)(17,18,22,26,23,25,21,19)(20,27,28,24,29,32,30,31);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19,22,25,23,26,21,18)(20,27,28,31,29,32,30,24);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,20,23,29)(18,24,25,31)(19,27,26,32)(21,30,22,28)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=h,f^2=g,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×#9284, #198856, #198858, … |
| maximal normal subgroup of | C2×#9284, #198856, #198858, … |
| maximal quotient of | #99932, #99934, #99944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2272);