Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19697 |
| automorphism | Aut(G) order 32768 |
| 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 | #3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
352 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 352 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(18,22)(19,26)(20,23)(24,27)(25,30)(28,31);(3,11)(7,15)(10,16)(12,13)(17,18,23,30,21,25,20,22)(19,27,31,29,28,24,26,32);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,19,23,31,21,28,20,26)(18,24,30,32,25,27,22,29);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(19,28)(24,27)(26,31)(29,32)Polycyclic presentation:
2^8|b^2=eg,c^2=fg,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=ch,d^c=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | #143189, #143191, #143192, … |
| maximal normal subgroup of | #143189, #143191, #143192, … |
| maximal quotient of | #191457, #206210, #206211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12700);