Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
257 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 257 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,4)(5,6,7,8)(9,10)(11,19)(12,24)(13,16)(14,27)(15,21)(17,20)(18,25)(22,31)(23,38)(26,39)(28,37)(29,40)(30,35)(32,34)(33,36);(6,8)(9,11,20,34,14,23,26,37,13,22,35,40,25,36,12,21)(10,15,24,33,18,29,30,31,16,28,39,38,27,32,17,19);(1,2,3,4)(10,16)(15,28)(17,30)(18,27)(19,31)(24,39)(29,32)(33,38)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eh,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #131859, #131889, #145026, … |
| maximal normal subgroup of | #131859, #131889, #145026, … |
| maximal quotient of | #140521, #140570, #140899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6192);