Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2308 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
691 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 691 conjugacy classes of subgroups — too many to draw. Only its 181 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) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,10,12,6)(4,9)(5,8)(7,11,16,13)(14,15);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14);(1,3)(2,7)(4,13)(5,12)(6,9)(8,16)(10,15)(11,14);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)Polycyclic presentation:
2^8|a^2=g,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=gh,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #151822, #151827, #151832, … |
| maximal normal subgroup of | #151822, #151827, #151832, … |
| maximal quotient of | #213768, #213772, #213774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14770);