Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
179 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 179 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:
(1,2,4,6,5,9,12,8)(3,10,13,7,11,14,16,15)(17,18,21,24,20,23,25,31)(19,22,27,30,26,29,28,32);(2,6,9,8)(3,11)(7,10,15,14)(13,16)(17,19,25,27,20,26,21,28)(18,22,24,32,23,29,31,30);(1,3,12,16,5,11,4,13)(2,7,6,14,9,15,8,10)(18,23)(19,26)(24,31)(27,28)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cg,d^b=dh,f^b=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #138838, #138839, #138850, … |
| maximal normal subgroup of | #138838, #138839, #138850, … |
| maximal quotient of | #102072, #102074, #102076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2780);