Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
501 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10,12,14)(11,15,16,13)(18,21)(19,23)(25,26)(27,31)(28,30)(29,32);(2,5)(6,8)(17,18)(19,22)(20,21)(23,24)(25,27,30,29)(26,31,28,32);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,21)(22,23)(25,28,30,26)(27,31,29,32);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,13,14,15)(17,19,20,23)(18,22,21,24)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^8|a^2=efg,b^2=fgh,c^2=g,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #144500, #144508, #145840, … |
| maximal normal subgroup of | #144500, #144508, #145840, … |
| maximal quotient of | #155759, #155818, #156010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10000);