Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3549 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
430 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 69 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 — 3 generators on 16 points:
(1,2)(3,11,13,15)(4,8,14,7)(5,9)(6,10)(12,16);(2,7)(4,14)(5,12)(8,10)(9,11)(15,16);(1,3)(2,7)(4,12)(5,14)(6,13)(8,10)(9,15)(11,16)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,e^b=eg,f^b=fgh,d^c=dgh,e^c=eh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152564, #152567, #152591, … |
| maximal normal subgroup of | #152564, #152567, #152591, … |
| maximal quotient of | #129585, #129589, #129627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5766);