Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×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 | |
179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,21,26,32,18,22,27,33,28,34,38,42)(15,19,24,30,25,31,37,41,23,29,35,39,36,40,43,44);(1,2,3,4)(6,9)(10,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,30)(21,31)(22,29)(26,37)(27,35)(28,36)(32,41)(33,39)(34,40)(38,43)(42,44);(5,7,8,11)(6,10,9,12)(15,23)(19,29)(24,35)(25,36)(30,39)(31,40)(37,43)(41,44)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=d,e^2=g,g^2=h,b^a=bd,c^b=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #94000, #126542, #126543, … |
| maximal normal subgroup of | #94000, #126542, #126543, … |
| maximal quotient of | #128205, #128209, #128210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5580);