Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9263 |
| automorphism | Aut(G) = C23×#2403 (order 1024) · Inn(G) = #386 · Out(G) = C24 |
| 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 | #121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 200 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:
(1,2,4,7,6,10,5,9)(3,11,12,8,13,15,14,16)(18,22,24,27)(19,25,26,28);(2,7,10,9)(8,15,16,11)(17,18,20,27,21,24,29,22)(19,23,28,31,26,32,25,30);(1,3)(2,8)(4,12)(5,14)(6,13)(7,15)(9,11)(10,16)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,29)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=egh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9266, #198542, #198543, … |
| maximal normal subgroup of | C2×#9266, #198542, #198543, … |
| maximal quotient of | #99917, #99918, #99919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2254);