Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
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 | |
273 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 61 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)(7,18)(8,24)(9,27)(10,16)(11,30)(12,17)(13,20)(14,23)(15,19)(21,33)(22,32)(25,31)(26,35)(28,34)(29,36);(2,4)(5,7,12,23)(6,13,17,31)(8,9,26,29)(10,22,11,21)(14,27,18,36)(15,33,34,32)(16,35,30,24)(19,20,28,25);(5,8,12,26)(6,14,17,18)(7,19,23,28)(9,21,29,22)(10,20,11,25)(13,15,31,34)(16,33,30,32)(24,36,35,27)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dg,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125649, #125650, #125723, … |
| maximal normal subgroup of | #125649, #125650, #125723, … |
| maximal quotient of | #142635, #142639, #142713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6321);