Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C42 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C24 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
293 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 175 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)(5,6,9,14,10,16,24,33,11,15,25,32,8,13,22,30)(7,17,20,34,21,12,26,28,18,29,23,36,19,27,35,31);(2,4)(5,7)(6,12,15,27)(8,19)(9,23)(10,21)(11,18)(13,17,16,29)(14,31,32,28)(20,25)(22,26)(24,35)(30,36,33,34);(7,18)(12,27)(17,29)(19,21)(20,23)(26,35)(28,31)(34,36)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=h,e^2=g,g^2=h,b^a=bd,f^a=fh,c^b=ch,e^b=eh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93437, #124166, #124337, … |
| maximal normal subgroup of | #93437, #124166, #124337, … |
| maximal quotient of | #126551, #126562, #126563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5535);