Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
453 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(2,7)(3,10)(4,5)(9,12)(11,13)(14,16)(18,23)(20,21)(24,31)(25,26)(27,29)(30,32);(1,2,6,9)(3,8,11,15)(4,12,5,7)(10,16,13,14)(17,18,22,26)(19,24,28,32)(20,23,21,25)(27,30,29,31);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,15)(12,16)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,31)(26,32);(2,9)(7,12)(8,15)(14,16)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30)Polycyclic presentation:
2^8|b^2=g,d^2=h,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh31 conjugacy classes — view the full character table.
| Fitting subgroup of | #19096⋊C3 |
| maximal subgroup of | #347035, #347037, #347038, … |
| maximal normal subgroup of | #347035, #347037, #347038, … |
| maximal quotient of | #168935, #168949, #169000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12084);