Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 262144 |
| 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 | |
561 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(3,7)(6,8)(10,13)(11,15)(17,18,21,19)(20,22,24,23)(27,31)(29,32);(2,5)(6,8)(10,13)(11,15)(18,19)(20,24)(25,26,28,30)(27,32,31,29);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,22,24,23)(25,27,28,31)(26,29,30,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,24)(18,22,19,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^8|a^2=e,b^2=egh,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | #125828, #125871, #126014, … |
| maximal normal subgroup of | #125828, #125871, #126014, … |
| maximal quotient of | #156591, #156599, #156649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10294);