Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 69 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)(5,6,9,14,24,33,25,34,11,16,8,13,10,15,26,35)(7,17,19,29,27,32,22,12,21,28,18,36,20,31,23,30);(1,2,3,4)(5,7)(6,12)(8,22)(9,23)(10,27)(11,21)(13,31)(14,32)(15,36)(16,30)(17,34)(18,25)(19,26)(20,24)(28,35)(29,33);(5,7)(6,12)(8,23)(9,22)(10,27)(11,21)(13,32)(14,31)(15,36)(16,30)(17,35)(18,26)(19,25)(20,24)(28,34)(29,33)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=gh,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=eg,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13026, #494534, #494535, … |
| maximal normal subgroup of | C2×#13026, #494534, #494535, … |
| maximal quotient of | #136696, #136730, #136900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6014);