Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 69 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)(5,6,11,18,31,32,30,16,13,20,10,17,12,19,9,29)(7,21,24,14,28,15,22,35,26,36,23,34,25,33,8,27);(5,7,12,28,13,26,31,25)(6,14,32,27,20,34,19,35)(8,9,24,10,22,30,23,11)(15,18,21,16,33,17,36,29);(2,4)(5,8)(6,15)(7,9)(10,28)(11,25)(12,24)(13,22)(14,18)(16,27)(17,34)(19,36)(20,33)(21,32)(23,31)(26,30)(29,35)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=eg,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13028, #494542, #494543, … |
| maximal normal subgroup of | C2×#13028, #494542, #494543, … |
| maximal quotient of | #136697, #136731, #136904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6016);