Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,10)(8,12,16,11)(9,13)(14,18)(15,17,20,19);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,17)(13,18)(16,20);(6,11)(7,14)(8,16)(10,17)(12,13)(18,19)Polycyclic presentation:
2^8|a^2=f,b^2=d,e^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=ch,e^c=eh,f^c=fgh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #130280, #130282, #130342, … |
| maximal normal subgroup of | #130280, #130282, #130342, … |
| maximal quotient of | #134232, #134257, #134286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5787);