Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) = C2×#72237540 (order 2048) · Inn(G) = #127 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1955 |
| prime radical | #525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
219 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,6,12,7,11,23,27,8,3,9,15,5,14,22,30)(4,16,20,31,21,13,28,29,18,17,24,32,19,10,25,26);(1,3,6,15,7,14,23,30,8,2,9,12,5,11,22,27)(4,17,24,31,21,10,25,29,18,16,20,32,19,13,28,26);(1,4,5,19,8,18,7,21)(2,10,11,16,3,13,14,17)(6,20,22,25,9,24,23,28)(12,26,27,31,15,29,30,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=e,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cg,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94381, #94382, #94397, … |
| maximal normal subgroup of | #94381, #94382, #94397, … |
| maximal quotient of | #140561, #140565, #140583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6188);