Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13609 |
| automorphism | Aut(G) = C2×#493397 (order 1024) · Inn(G) = #2127 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4⋊Q8 | C23 |
| prime radical | #3221 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
158 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,10,6,11,17,25)(3,8,13,21,12,22,26,32)(5,7,15,20,18,9,16,24)(14,19,29,30,27,23,28,31);(2,7)(3,12)(4,15)(5,18)(8,19)(9,11)(13,28)(16,17)(20,24)(22,23)(26,29)(30,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,23)(10,21)(11,22)(15,29)(16,28)(17,26)(18,27)(20,30)(24,31)(25,32)Polycyclic presentation:
2^8|a^2=e,d^2=gh,e^2=h,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=cg,d^b=dgh,e^b=efg,f^b=fh,d^c=dh,e^d=egh,f^e=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #152139, #152141, #152142, … |
| maximal normal subgroup of | #152139, #152141, #152142, … |
| maximal quotient of | #149595, #149638, #149715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6599);