Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
654 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 139 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:
(5,6,12,18)(7,19,9,27)(8,23,26,32)(10,17,11,16)(13,22,15,21)(14,28,31,25)(20,30,24,29)(33,35,36,34);(1,2)(3,4)(5,7,11,22,12,9,10,21)(6,13,17,27,18,15,16,19)(8,20,28,35,26,24,25,34)(14,29,32,36,31,30,23,33);(2,4)(5,8)(6,14)(7,20)(9,24)(10,25)(11,28)(12,26)(13,29)(15,30)(16,23)(17,32)(18,31)(19,33)(21,34)(22,35)(27,36);(5,9,11,21,12,7,10,22)(6,15,17,19,18,13,16,27)(8,24,28,34,26,20,25,35)(14,30,32,33,31,29,23,36)Polycyclic presentation:
2^8|a^2=h,b^2=eh,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,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139569, #149947, #150464, … |
| maximal normal subgroup of | #139569, #149947, #150464, … |
| maximal quotient of | #262679, #262683, #262688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17270);