Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
572 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 195 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 — 4 generators on 16 points:
(2,6,7,8)(9,10,13,16,14,11,12,15);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,15,11,16);(9,12,14,13)(10,15,11,16)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=g,e^2=h,f^2=g,b^a=be,e^a=eg,e^b=eh,f^b=fgh76 conjugacy classes — view the full character table.
| maximal subgroup of | #107700, #107715, #107994, … |
| maximal normal subgroup of | #107700, #107715, #107994, … |
| maximal quotient of | #160424, #160432, #160434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10626);