Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7262 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #340 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C23×C8 | C22 |
| prime radical | C24 | Q16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 51 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 — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,18,15,20,14,19)(11,21,23,38,25,39,24,35)(12,26,28,37,30,36,29,17)(16,31,32,22,34,27,33,40);(2,5,7,6)(9,11,13,24,15,25,14,23)(10,16,18,33,20,34,19,32)(12,27,28,22,30,31,29,40)(17,35,26,39,37,38,36,21)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,c^b=cg,d^b=dh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7265, #127528, #127559, … |
| maximal normal subgroup of | C2×#7265, #127528, #127559, … |
| maximal quotient of | #93052, #93053, #93056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 253);