Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 167 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,14,9,26,25,36,12,17,24,34,11,16,8,15)(7,18,22,13,28,31,27,30,23,29,19,33,21,35,20,32);(1,2,3,4)(5,7,9,21,12,23,11,28)(6,13,16,30,17,33,26,32)(8,19,10,27,25,22,24,20)(14,29,15,35,34,18,36,31);(5,8,11,24,12,25,9,10)(6,14,26,36,17,34,16,15)(7,19,28,20,23,22,21,27)(13,29,32,31,33,18,30,35)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=eh,d^2=g,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dg,f^b=fgh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #140261, #140263, #140300, … |
| maximal normal subgroup of | #140261, #140263, #140300, … |
| maximal quotient of | #141848, #141868, #142079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6261);