Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | Aut(G) = C2×#512862 (order 1024) · Inn(G) = #2196 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2196 |
| commutator, Frattini, cosocle | #127 | C23 |
| prime radical | #3291 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
197 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 29 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:
(1,2,6,10)(3,7,13,21)(4,8,16,24)(5,17,18,9)(11,25,28,31)(12,20,29,30)(14,22,15,23)(19,27,32,26);(1,3,6,13)(2,7,10,21)(4,11,16,28)(5,12,18,29)(8,19,24,32)(9,20,17,30)(14,26,15,27)(22,25,23,31);(1,4)(2,8)(3,11)(5,14)(6,16)(7,19)(9,22)(10,24)(12,26)(13,28)(15,18)(17,23)(20,25)(21,32)(27,29)(30,31)Polycyclic presentation:
2^8|a^2=h,b^2=h,f^2=h,b^a=bd,c^a=ce,f^a=fg,e^b=efgh,g^b=gh,d^c=df,f^c=fh,g^c=gh,e^d=egh,f^d=fh,f^e=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #153626, #153628, #153630, … |
| maximal normal subgroup of | #153626, #153628, #153630, … |
| maximal quotient of | #151145, #151161, #151191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6664);