Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | Aut(G) = C2×#72236702 (order 2048) · Inn(G) = #2044 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2044 |
| commutator, Frattini, cosocle | C2×#41 | C23 |
| prime radical | #535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 212 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:
(2,7)(3,11)(4,15)(5,18)(8,22)(9,24)(10,16)(12,28)(13,30)(14,27)(19,21)(20,23)(25,29)(26,31);(1,2,6,10)(3,12,14,8)(4,16,15,7)(5,19,18,9)(11,25,27,23)(13,20,31,29)(17,21,32,24)(22,30,28,26);(1,3,4,13,6,14,15,31)(2,8,7,20,10,12,16,29)(5,11,17,26,18,27,32,30)(9,23,21,22,19,25,24,28)Polycyclic presentation:
2^8|b^2=h,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,e^b=eg,f^c=fh,g^c=gh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13433, #499150, #499151, … |
| maximal normal subgroup of | C2×#13433, #499150, #499151, … |
| maximal quotient of | #144671, #144676, #144707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6421);