Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
198 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(1,2)(3,4)(5,6)(7,19)(8,26)(9,16)(10,15)(11,28)(12,31)(13,18)(14,21)(17,29)(20,22)(23,35)(24,27)(25,36)(30,34)(32,33);(5,7,13,25)(6,14,18,35)(8,27,15,33)(9,22,11,30)(10,21,26,23)(12,32,29,24)(16,19,28,36)(17,20,31,34);(2,4)(5,8,9,17,29,16,26,6,13,15,11,31,12,28,10,18)(7,20,21,27,32,35,30,36,25,34,23,33,24,14,22,19)Polycyclic presentation:
2^8|b^2=h,c^2=d,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,d^b=dg,e^b=egh,f^b=fg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13446, #499182, #499184, … |
| maximal normal subgroup of | C2×#13446, #499182, #499184, … |
| maximal quotient of | #145366, #145369, #145370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6434);