Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10280 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 264 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,32)(30,31);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(20,29)(22,31)(23,28)(26,30);(4,14)(6,15)(7,12)(10,13)(17,19,21,27)(18,22,24,31)(20,30,29,26)(23,32,28,25)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=h,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eh,d^c=dg,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #126253, C2×#10297, #234132, … |
| maximal normal subgroup of | #126253, C2×#10297, #234132, … |
| maximal quotient of | #102982, #102992, #103153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3285);