Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
177 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,21,26,36)(12,27)(13,17)(14,31)(15,20)(16,24,34,29)(18,22)(19,28)(23,33,38,40)(25,32,39,37)(30,35);(2,5)(6,7)(9,11,22,38,28,39,13,29,15,26,12,23,14,25,30,24)(10,16,17,32,31,40,18,36,20,34,35,37,19,33,27,21);(1,3,4,8)(2,6,5,7)(9,12,28,30,15,22,14,13)(10,17,31,18,20,35,19,27)(11,23,39,24,26,38,25,29)(16,32,40,36,34,37,33,21)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=eh,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131923, #131924, #131925, … |
| maximal normal subgroup of | #131923, #131924, #131925, … |
| maximal quotient of | #141888, #141890, #141915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6276);