Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #476 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1948 |
| prime radical | C23 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 140 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:
(1,2,6,13,8,14,7,12)(3,15,19,29,20,26,18,32)(4,21,22,10,23,11,5,24)(9,25,27,17,28,31,30,16);(1,3)(2,9,14,28)(4,16)(5,25)(6,18)(7,19)(8,20)(10,26,24,15)(11,29,21,32)(12,27,13,30)(17,23)(22,31);(1,4,6,22,8,23,7,5)(2,10,12,21,14,24,13,11)(3,16,18,31,20,17,19,25)(9,26,27,32,28,15,30,29)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=e,d^2=g,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136363, #136365, #136421, … |
| maximal normal subgroup of | #136363, #136365, #136421, … |
| maximal quotient of | #140390, #140391, #140405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6176);