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) = 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 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,22)(21,30)(23,25)(26,27)(28,29)(31,32);(2,5)(6,8)(10,13)(14,15)(17,19,22,28)(18,24,20,29)(21,27,23,31)(25,32,30,26);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,18)(19,26,28,32)(21,30,23,25)(24,31,29,27);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(17,21,22,23)(18,25,20,30)(19,27,28,31)(24,32,29,26)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=ef,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×#30140, #6434209, #6434217, … |
| maximal normal subgroup of | C2×#30140, #6434209, #6434217, … |
| maximal quotient of | #380422, #380423, #380478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23128);