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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2196 |
| commutator, Frattini, cosocle | #127 | C23 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
227 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 227 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(4,12)(6,13)(14,15);(1,2,5,8)(3,6,11,15)(4,7,12,16)(9,13,10,14);(1,3)(2,6)(4,9)(5,11)(7,13)(8,15)(10,12)(14,16)Polycyclic presentation:
2^8|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 | #153625, #153627, #153629, … |
| maximal normal subgroup of | #153625, #153627, #153629, … |
| maximal quotient of | #151143, #151159, #151189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6662);