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) = C23 |
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 | |
319 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,26,22,25,20,21)(19,27,28,32,30,24,31,29);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,31)(21,32)(22,30)(23,28)(25,27)(26,29);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20,22,23)(18,21,25,26)(19,28,30,31)(24,29,27,32);(1,4,5,3)(2,8,6,7)(17,21,20,25,22,26,23,18)(19,29,31,24,30,32,28,27)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=e,d^2=eg,g^2=h,b^a=be,c^a=cf,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21678, #6373590, #6373593, … |
| maximal normal subgroup of | C2×#21678, #6373590, #6373593, … |
| maximal quotient of | #354522, #354523, #354526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14666);