Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13584 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#353 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #127 | C23 |
| prime radical | #3151 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(1,2,4,9)(3,8,11,15)(5,7,12,14)(6,10,13,16);(2,7)(3,5)(4,12)(8,10)(11,13)(14,15);(1,3,6,5)(2,8,10,7)(4,11,13,12)(9,15,16,14)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=gh,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=dgh,e^b=efg,f^b=fh,e^d=egh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94791, #152051, #152089, … |
| maximal normal subgroup of | #94791, #152051, #152089, … |
| maximal quotient of | #149575, #149625, #149694, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6573);