Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#472 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,4,3)(5,6,10,16)(7,12,20,28)(8,21,24,30)(9,15,22,14)(11,19,26,18)(13,29,31,23)(17,32,35,25)(27,34,36,33);(1,3,4,2)(5,7,9,18,10,20,22,19)(6,11,14,12,16,26,15,28)(8,17,23,33,24,35,29,34)(13,25,21,27,31,32,30,36);(2,3)(5,7)(6,12)(8,17)(9,18)(10,20)(11,15)(13,27)(14,26)(16,28)(19,22)(21,32)(23,33)(24,35)(25,30)(29,34)(31,36);(5,8,22,29,10,24,9,23)(6,13,15,30,16,31,14,21)(7,17,19,34,20,35,18,33)(11,25,28,36,26,32,12,27)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #502187, #502192, #502211, … |
| maximal normal subgroup of | #502187, #502192, #502211, … |
| maximal quotient of | #288030, #288035, #288044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18421);