Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C23 |
| commutator | C22 | C2×C4×C8 |
| socle | C23 | C2×C42 |
| prime radical | C24 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
367 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(2,4)(5,6,7,9,8,10,11,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=g,f^2=h,b^a=bd,c^b=cg112 conjugacy classes — view the full character table.
| maximal subgroup of | #92964, #92968, #92972, … |
| maximal normal subgroup of | #92964, #92968, #92972, … |
| maximal quotient of | #94868, #94869, #94870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 837);