Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
221 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 71 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)(5,6,11,17,13,18,12,21)(7,19,22,29,15,9,16,10)(8,23,26,20,28,35,27,31)(14,30,32,36,33,24,34,25);(1,2,3,4)(5,7,19,21,12,22,10,6,13,15,9,17,11,16,29,18)(8,20,30,34,27,35,25,14,28,31,24,32,26,23,36,33);(5,8,9,24,12,27,29,36,13,28,19,30,11,26,10,25)(6,14,7,20,17,32,22,35,18,33,15,31,21,34,16,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=eg,d^2=g,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152526, #152527, #152531, … |
| maximal normal subgroup of | #152526, #152527, #152531, … |
| maximal quotient of | #140729, #140774, #141208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6233);