Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
291 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,15,12,11,14,16);(1,3)(2,5,6,7)(4,8)(10,11)(15,16);(9,11,13,16,12,10,14,15)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,e^2=h,b^a=bd,f^a=fg,c^b=ch,d^b=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93000, #93001, #93004, … |
| maximal normal subgroup of | #93000, #93001, #93004, … |
| maximal quotient of | #96672, #96673, #96674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1148);