Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| 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 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,8,11,22)(7,16,19,33,20,32,28,34)(9,23,25,21,27,36,26,14)(13,29,30,18,17,35,31,24);(5,7,9,24,11,19,26,29,12,20,27,18,10,28,25,35)(6,13,21,16,15,31,36,34,8,17,14,32,22,30,23,33);(2,4)(5,8)(6,12)(7,17)(9,14)(10,15)(11,22)(13,20)(16,18)(19,30)(21,27)(23,26)(24,32)(25,36)(28,31)(29,33)(34,35)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13072, #494658, #494659, … |
| maximal normal subgroup of | C2×#13072, #494658, #494659, … |
| maximal quotient of | #137617, #137657, #137982, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6060);