Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator | C22 | C2×C4×C8 |
| socle, prime radical | C23 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,4)(5,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,26)(20,27)(21,25)(22,30)(23,28)(24,29)(31,37)(32,35)(33,36)(34,38)(39,40);(2,4)(9,11,12,19,13,20,22,31,14,21,23,32,24,33,34,39)(10,15,17,26,18,27,30,37,16,25,28,35,29,36,38,40);(5,7,8,6)(10,16)(15,25)(17,28)(18,29)(26,35)(27,36)(30,38)(37,40)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,f^2=g,g^2=h,b^a=bd,c^a=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #93355, #134392, #134417, … |
| maximal normal subgroup of | #93355, #134392, #134417, … |
| maximal quotient of | #129037, #129046, #129065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5642);