Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 337 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,20)(11,21,24,17)(12,25,27,39)(13,29,15,19)(16,30,32,23)(18,28,34,26)(22,36,38,31)(33,37,35,40);(9,11,13,23,14,24,15,30)(10,16,19,21,20,32,29,17)(12,22,26,37,27,38,28,40)(18,31,25,35,34,36,39,33);(1,3,4,8)(2,5,6,7)(9,11,14,24)(10,17,20,21)(12,22,27,38)(13,23,15,30)(16,29,32,19)(18,33,34,35)(25,31,39,36)(26,37,28,40);(9,12,15,28,14,27,13,26)(10,18,29,39,20,34,19,25)(11,22,30,40,24,38,23,37)(16,31,17,33,32,36,21,35)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=f,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 | #503647, #503648, #503650, … |
| maximal normal subgroup of | #503647, #503648, #503650, … |
| maximal quotient of | #305773, #305776, #305781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19070);