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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #127 | C23 |
| prime radical | #3220 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #535 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 174 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)(3,12,14,7)(4,8)(5,10,20,9)(6,11)(13,27,28,21)(15,22,19,23)(16,29,30,32)(17,25)(18,24,31,26);(1,3)(2,7)(4,16)(5,19)(6,15)(8,21)(9,22)(10,12)(11,23)(13,17)(14,20)(18,28)(24,27)(25,32)(26,29)(30,31);(1,4)(2,8)(3,13)(5,18)(6,17)(7,21)(9,26)(10,24)(11,25)(12,27)(14,28)(15,16)(19,30)(20,31)(22,29)(23,32)Polycyclic presentation:
2^8|a^2=e,d^2=gh,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 | #152133, #152135, #152136, … |
| maximal normal subgroup of | #152133, #152135, #152136, … |
| maximal quotient of | #149595, #149637, #149715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6597);