Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13677 |
| automorphism | Aut(G) = C22×#2196 (order 512) · Inn(G) = #2196 · Out(G) = C22 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2196 |
| commutator, Frattini, cosocle | #124 | C23 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 200 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:
(2,7)(4,15)(8,24)(9,23)(10,27)(11,16)(12,29)(13,19)(18,20)(22,25)(26,30)(28,32);(1,2)(3,8)(4,16)(5,10)(6,11)(7,15)(9,21)(12,30)(13,25)(14,26)(17,32)(18,27)(19,28)(20,23)(22,31)(24,29);(1,3,6,14)(2,8,11,26)(4,17,15,31)(5,19,21,13)(7,22,16,32)(9,25,10,28)(12,20,29,18)(23,24,27,30)Polycyclic presentation:
2^8|c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fg,d^b=dh,e^b=efgh,g^b=gh,d^c=df,f^c=fh,g^c=gh,e^d=egh,f^d=fh,f^e=fh22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13678, #512875, #512877, … |
| maximal normal subgroup of | C2×#13678, #512875, #512877, … |
| maximal quotient of | #151143, #151147, #151163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6666);