Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = #120 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
237 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 237 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) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(6,9,10,12)(13,14,17,21,19,22,18,27)(15,23,24,28,16,25,26,20);(1,2,3,4)(7,11)(9,12)(13,15,19,16)(14,20,22,28)(17,26,18,24)(21,25,27,23);(5,6)(7,12)(8,10)(9,11)(13,16,19,15)(14,20,22,28)(17,26,18,24)(21,23,27,25)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9000, #194474, #194477, … |
| maximal normal subgroup of | C2×#9000, #194474, #194477, … |
| maximal quotient of | #99536, #99542, #99632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1988);