Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3155 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(2,3)(5,6)(7,12)(8,13)(9,26)(10,27)(11,16)(14,23)(15,24)(17,35)(18,31)(19,36)(20,29)(21,33)(22,30)(25,34)(28,32);(1,2)(3,4)(5,7)(6,12)(8,22)(9,17)(10,28)(11,21)(13,30)(14,31)(15,29)(16,33)(18,23)(19,25)(20,24)(26,35)(27,32)(34,36);(1,3,4,2)(6,13)(7,17)(8,23)(9,25)(10,24)(12,29)(14,15)(16,34)(18,21)(19,28)(20,22)(26,27)(31,35)(32,33)Polycyclic presentation:
2^8|c^2=d,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,e^b=eg,f^b=fgh,g^b=gh,e^c=eg,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13411, #499085, #499086, … |
| maximal normal subgroup of | C2×#13411, #499085, #499086, … |
| maximal quotient of | #144889, #144890, #144928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6399);