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) = C22 |
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 | |
206 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 206 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:
(1,2,4,3)(6,9)(7,11)(10,12)(13,14,19,24)(15,25,16,27)(17,23,18,22)(20,28,21,26);(1,3,4,2)(7,11)(9,12)(13,15,19,16)(14,20,24,21)(17,28,18,26)(22,27,23,25);(5,6)(7,12)(8,10)(9,11)(13,16,19,15)(14,21,24,20)(17,28,18,26)(22,27,23,25)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dg,e^b=eh,d^c=dh,e^c=egh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8912, #193018, #193021, … |
| maximal normal subgroup of | C2×#8912, #193018, #193021, … |
| maximal quotient of | #99327, #99333, #99359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1900);