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 | #3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 115 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,25)(13,29)(14,20)(15,19)(16,23)(17,24)(18,26)(22,36)(27,39)(28,34)(30,32)(31,38)(33,40)(35,37);(1,3,4,7)(2,5,6,8)(9,11,15,30,14,24,13,23)(10,16,29,17,20,32,19,21)(12,22,28,40,27,38,26,37)(18,31,39,33,34,36,25,35);(2,6)(5,8)(9,11)(10,17)(12,22)(13,23)(14,24)(15,30)(16,19)(18,33)(20,21)(25,31)(26,37)(27,38)(28,40)(29,32)(34,35)(36,39);(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|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 | #501849, #501850, #501852, … |
| maximal normal subgroup of | #501849, #501850, #501852, … |
| maximal quotient of | #280989, #280990, #280995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18040);