Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10176 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
232 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(18,23)(19,26)(20,21)(25,30)(27,31)(28,29);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,18,22,25)(19,27,29,31)(20,30,21,23)(24,26,32,28);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19,22,29)(18,24,25,32)(20,26,21,28)(23,31,30,27)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,f^b=fh,d^c=dh,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10191, #237204, #237208, … |
| maximal normal subgroup of | C2×#10191, #237204, #237208, … |
| maximal quotient of | #102498, #102500, #102503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3179);