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)(7,18,20,29,21,15,9,13)(8,22,24,32,25,31,27,14)(19,28,23,30,34,36,35,33);(1,2,3,4)(5,7,11,20,12,21,10,9)(6,13,26,15,17,29,16,18)(8,19,27,35,25,34,24,23)(14,28,22,30,32,36,31,33);(2,4)(5,8)(6,14)(7,19)(9,23)(10,24)(11,27)(12,25)(13,28)(15,30)(16,31)(17,32)(18,33)(20,35)(21,34)(22,26)(29,36);(5,9,10,21,12,20,11,7)(6,15,16,13,17,18,26,29)(8,23,24,34,25,35,27,19)(14,30,31,28,32,33,22,36)Polycyclic presentation:
2^8|a^2=f,b^2=egh,d^2=e,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 | #139402, #139405, #143212, … |
| maximal normal subgroup of | #139402, #139405, #143212, … |
| maximal quotient of | #321510, #321515, #321519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21483);