Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #95 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C43 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
139 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 55 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 — 2 generators on 32 points:
(1,2,5,8)(3,6,7,4)(9,10,12,14,13,16,11,15)(17,18,20,25,22,26,21,24)(19,27,28,23,30,31,29,32);(1,3,6,2,5,7,4,8)(10,14)(11,12)(15,16)(17,19,21,29,22,30,20,28)(18,23,25,31,26,32,24,27)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=fg,e^2=f,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7041, #95999, #96000, … |
| maximal normal subgroup of | C2×#7041, #95999, #96000, … |
| maximal quotient of | #92809, #92811, #93608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29);