Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12948 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1864 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1864 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
197 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,6,13,8,14,7,25)(3,15,17,29,19,26,18,30)(4,20,23,12,5,9,21,11)(10,22,27,16,28,24,31,32);(2,9,14,20)(4,21)(5,23)(6,7)(10,26,28,15)(11,13,12,25)(16,22)(17,18)(24,32)(27,29,31,30);(1,3)(2,10)(4,22)(5,24)(6,18)(7,17)(8,19)(9,26)(11,29)(12,30)(13,31)(14,28)(15,20)(16,21)(23,32)(25,27)Polycyclic presentation:
2^8|a^2=f,b^2=fg,d^2=g,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dg,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12949, #494321, #495739, … |
| maximal normal subgroup of | C2×#12949, #494321, #495739, … |
| maximal quotient of | #135904, #135928, #136223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5937);