Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33618 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#466 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 117 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:
(2,8)(3,14)(4,9)(5,6)(10,25)(11,21)(12,23)(13,22)(15,29)(16,19)(17,18)(20,27)(26,32)(28,31);(1,2,9,23,6,13,18,21,7,11,17,22,5,12,4,8)(3,10,24,25,14,27,29,31,16,26,30,32,19,28,15,20);(1,3)(2,10)(4,15)(5,19)(6,14)(7,16)(8,20)(9,24)(11,26)(12,28)(13,27)(17,30)(18,29)(21,31)(22,32)(23,25);(2,11)(8,21)(10,26)(12,13)(20,31)(22,23)(25,32)(27,28)Polycyclic presentation:
2^8|b^2=eg,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gh,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #495899, #498379, #501062, … |
| maximal normal subgroup of | #495899, #498379, #501062, … |
| maximal quotient of | #490472, #490479, #490544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26606);