Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13502 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #2077 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2077 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
138 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,7,22,21,12,4,11,6,10,24,9,5,13,20,28)(3,14,31,32,17,26,16,29,18,8,23,27,15,30,19,25);(1,3)(2,8)(4,19)(5,15)(6,18)(7,23)(9,27)(10,14)(11,29)(12,30)(13,26)(16,20)(17,21)(22,32)(24,31)(25,28);(2,9,10,22)(3,15)(4,7)(5,21)(8,25,14,29)(11,13,28,12)(16,19)(17,18)(20,24)(26,32,30,27)Polycyclic presentation:
2^8|a^2=eg,c^2=d,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,e^c=eg,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13502, #499390, #499514, … |
| maximal normal subgroup of | C2×#13502, #499390, #499514, … |
| maximal quotient of | #146042, #146046, #146072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6490);