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) = C23 |
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 | |
252 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 252 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:
(1,2,5,9)(3,7,11,16)(4,6,12,8)(10,15,13,14)(17,18)(19,22)(20,23)(21,25)(24,29)(26,30)(27,32)(28,31);(2,6)(4,12)(7,15)(8,9)(10,13)(14,16)(17,19,21,27)(18,22,25,32)(20,28,29,26)(23,31,24,30);(1,3,5,11)(2,7,9,16)(4,13,12,10)(6,14,8,15)(18,23)(20,29)(22,30)(24,25)(26,28)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,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 | #126254, C2×#10295, #234127, … |
| maximal normal subgroup of | #126254, C2×#10295, #234127, … |
| maximal quotient of | #102980, #102990, #103143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3283);