Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
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 | |
295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,23)(21,24)(22,27)(26,28)(29,30)(31,32);(2,5)(6,8)(17,19,21,28,20,27,29,31)(18,22,24,32,23,25,30,26);(3,7)(6,8)(9,11,12,16)(10,13,14,15)(17,18)(19,26)(20,23)(21,30)(22,31)(24,29)(25,28)(27,32);(1,3,4,7)(2,6,5,8)(18,23)(22,25)(24,30)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30315, #6435986, #6435994, … |
| maximal normal subgroup of | C2×#30315, #6435986, #6435994, … |
| maximal quotient of | #380819, #380820, #380851, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23303);