Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13681 |
| 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 | C4⋊Q8 | C23 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
192 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 192 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:
(2,8)(3,12)(4,15)(5,18)(10,28)(11,16)(13,32)(14,20)(17,22)(19,30)(23,25)(24,31);(1,2,7,11)(3,9,14,27)(4,16,15,8)(5,19,18,28)(6,10,21,30)(12,31,20,24)(13,26,22,29)(17,25,32,23);(1,3)(2,9)(4,17)(5,20)(6,22)(7,14)(8,23)(10,29)(11,27)(12,18)(13,21)(15,32)(16,25)(19,24)(26,30)(28,31)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fg,d^b=dh,e^b=efgh,g^b=gh,d^c=df,e^c=eh,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 | #153619, #153620, #153622, … |
| maximal normal subgroup of | #153619, #153620, #153622, … |
| maximal quotient of | #151147, #151159, #151192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6670);