Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19733 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
356 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 115 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 — 4 generators on 32 points:
(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(2,8)(3,9)(6,13)(10,15)(18,23)(22,29)(24,30)(28,32)Polycyclic presentation:
2^8|b^2=e,c^2=f,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,d^c=dg40 conjugacy classes — view the full character table.
| Fitting subgroup of | #19733⋊C3 |
| maximal subgroup of | C2×#19733, #6357385, #6357427, … |
| maximal normal subgroup of | C2×#19733, #6357385, #6357427, … |
| maximal quotient of | #191527, #206275, #206286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12721);