Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) = C25×D42 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #385 |
| prime radical | C2×#41 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
186 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,13)(3,10,11,7,12,15,14,16)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,3,5,11,6,12,4,14)(2,7,8,15,9,16,13,10)(20,29)(23,28)(25,32)(27,30);(17,19,21,26)(18,22,24,31)(20,30,29,27)(23,32,28,25)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eh,d^c=dh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8901, #192850, #192854, … |
| maximal normal subgroup of | C2×#8901, #192850, #192854, … |
| maximal quotient of | #99415, #99418, #99420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1889);