Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1874 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1874 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,6,11,7,13,22,25,8,14,4,20,5,12,21,9)(3,15,18,24,17,28,32,29,19,30,16,31,23,10,26,27);(2,9,14,25)(4,21)(5,7)(6,22)(10,24,28,31)(11,13,20,12)(15,27,30,29)(16,26)(17,23)(18,32);(1,3)(2,10)(4,18)(5,23)(6,16)(7,17)(8,19)(9,24)(11,29)(12,30)(13,15)(14,28)(20,27)(21,32)(22,26)(25,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,e^b=eh,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13005, #494477, #495739, … |
| maximal normal subgroup of | C2×#13005, #494477, #495739, … |
| maximal quotient of | #136615, #136616, #136643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5993);