Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
511 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,17,10,26)(7,15,9,25,20,32,19,13)(8,21,23,31,24,30,27,14)(18,28,33,35,34,36,22,29);(1,2,3,4)(5,7,10,19,12,20,11,9)(6,13,16,15,17,25,26,32)(8,18,23,33,24,34,27,22)(14,28,30,29,31,36,21,35);(2,4)(5,8)(6,14)(7,18)(9,22)(10,23)(11,27)(12,24)(13,28)(15,29)(16,30)(17,31)(19,33)(20,34)(21,26)(25,36)(32,35);(5,9,11,20,12,19,10,7)(6,15,26,13,17,32,16,25)(8,22,27,34,24,33,23,18)(14,29,21,28,31,35,30,36)Polycyclic presentation:
2^8|a^2=f,b^2=eg,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #143817, #143818, #143821, … |
| maximal normal subgroup of | #143817, #143818, #143821, … |
| maximal quotient of | #321511, #321516, #321523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21484);