Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C22×C42 | C22 |
| socle | C23 | #95 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
224 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,19,21,20,22,23,24);(1,2,4,7,5,8,3,6)(9,11)(10,13,14,15)(12,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=h,e^2=f,b^a=bc,c^a=cf,e^a=efg,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7029, #96467, #96468, … |
| maximal normal subgroup of | C2×#7029, #96467, #96468, … |
| maximal quotient of | #92809, #92812, #92814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17);