Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(5,6,10,15,11,16,9,14)(7,17,20,12,21,27,19,28)(8,22,23,13,25,29,24,30)(18,32,33,36,34,26,35,31);(1,2)(3,4)(5,7,9,20,11,21,10,19)(6,12,14,27,16,28,15,17)(8,18,24,33,25,34,23,35)(13,26,22,31,30,32,29,36);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,29,16,30,15,22)(7,18,19,33,21,34,20,35)(12,26,17,31,28,32,27,36)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93952, #136156, #136157, … |
| maximal normal subgroup of | #93952, #136156, #136157, … |
| maximal quotient of | #120626, #120629, #120630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4285);