Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
279 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 279 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,29,33)(27,34,37,31)(28,38,30,32)(35,40,36,39);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,29,37)(26,31,33,34)(28,36,30,35)(32,39,38,40);(2,5)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(18,24)(21,23)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=eh,d^2=fh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30102, #6434794, #6434799, … |
| maximal normal subgroup of | C2×#30102, #6434794, #6434799, … |
| maximal quotient of | #380503, #380505, #381208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23090);