Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3155 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
515 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 515 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:
(1,2,3,4)(5,6)(7,13)(8,20)(9,12)(10,15)(11,16)(14,21)(17,25)(18,32)(19,27)(22,30)(23,26)(24,29)(28,31)(33,35)(34,36);(6,12,16,15)(7,17,19,24)(13,27)(14,26,30,20)(18,31,33,34)(25,29)(28,36)(32,35);(2,4)(5,7)(6,13)(8,18)(9,17)(10,24)(11,19)(12,25)(14,28)(15,29)(16,27)(20,32)(21,31)(22,34)(23,33)(26,35)(30,36);(5,8,9,21,11,23,10,22)(6,14,15,20,16,30,12,26)(7,18,17,31,19,33,24,34)(13,28,29,32,27,36,25,35)Polycyclic presentation:
2^8|a^2=f,b^2=eg,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #501625, #501630, #501631, … |
| maximal normal subgroup of | #501625, #501630, #501631, … |
| maximal quotient of | #321901, #321906, #321910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21489);