Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#472 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,16)(5,17)(9,24)(10,21)(14,15)(18,29)(19,23)(22,30)(25,27)(26,32);(1,2,5,7,6,10,17,21)(3,12,11,19,15,8,14,23)(4,9,16,20,18,24,29,31)(13,26,25,30,28,22,27,32);(1,3)(2,8)(4,13)(5,14)(6,15)(7,19)(9,22)(10,12)(11,17)(16,27)(18,28)(20,30)(21,23)(24,26)(25,29)(31,32);(1,4,17,29,6,18,5,16)(2,9,21,31,10,24,7,20)(3,13,11,25,15,28,14,27)(8,22,23,32,12,26,19,30)Polycyclic presentation:
2^8|b^2=f,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501858, #501859, #501864, … |
| maximal normal subgroup of | #501858, #501859, #501864, … |
| maximal quotient of | #280844, #280848, #280850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17976);