Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #367 |
| commutator | C16 | C42 |
| Frattini, cosocle | C22×C16 | C22 |
| prime radical | C23 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(6,12,13,28)(7,17)(8,22)(9,25)(10,19)(14,33,32,15)(16,36,29,31)(18,26)(20,24)(21,23)(27,34,35,30);(2,4)(5,6,8,14,9,15,7,12,19,29,17,30,18,27,22,36,11,13,24,32,26,33,21,28,10,16,23,34,25,35,20,31)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,e^2=f,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7474, #151774, #151969, … |
| maximal normal subgroup of | C2×#7474, #151774, #151969, … |
| maximal quotient of | #94023, #94046, #94047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 462);