Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| 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.
| 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)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,30)(28,33)(29,35)(31,38)(32,37)(34,36)(39,40);(2,5)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(18,24)(21,23)(26,31)(30,38)(32,40)(37,39)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=e,d^2=f,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×#30100, #6434794, #6434799, … |
| maximal normal subgroup of | C2×#30100, #6434794, #6434799, … |
| maximal quotient of | #380500, #380502, #380504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23088);