Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 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)(11,14)(12,16)(13,15)(17,18,22,26,21,25,27,31)(19,23,24,32,28,20,29,30);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,27,29,21,28,22,24)(18,23,31,30,25,20,26,32);(9,12,13,11)(10,14,15,16)(17,20,21,23)(18,24,25,29)(19,26,28,31)(22,30,27,32);(1,3,4,7)(2,6,5,8)(19,28)(20,23)(24,29)(30,32)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^b=dh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21348, #6372253, #6372256, … |
| maximal normal subgroup of | C2×#21348, #6372253, #6372256, … |
| maximal quotient of | #353733, #354010, #354011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14336);