Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
227 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 95 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:
(2,4)(5,6,8,13,9,14,19,26,10,12,18,24,20,25,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,15)(12,21)(13,22)(14,23)(18,27)(19,29)(20,28)(24,31)(25,32)(26,33)(30,35)(34,36);(6,12)(7,15)(8,18)(14,25)(17,28)(19,30)(22,31)(33,36)Polycyclic presentation:
2^8|a^2=f,b^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=ch,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93369, #93498, #93499, … |
| maximal normal subgroup of | #93369, #93498, #93499, … |
| maximal quotient of | #134219, #134247, #134272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5774);