Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
211 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14)(7,17)(8,23,10,26)(11,28,24,16)(12,20)(13,27,15,25)(18,33)(19,32)(21,36)(22,30)(29,34)(31,35);(2,4)(5,7,10,20,11,21,27,34,9,22,8,19,24,18,25,35)(6,12,15,30,16,31,26,36,14,32,13,17,28,29,23,33);(7,18,22,21)(8,10)(12,29,32,31)(13,15)(17,33,30,36)(19,35,20,34)(23,26)(25,27)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eh,d^2=g,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,c^b=cg,e^b=eh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12648, #493077, #493078, … |
| maximal normal subgroup of | C2×#12648, #493077, #493078, … |
| maximal quotient of | #128556, #128573, #128606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5636);