Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #106 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #334 |
| prime radical | C23 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
101 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)(5,6,9,15,11,16,26,32,12,17,25,14,10,28,8,24)(7,18,20,29,22,33,19,30,23,31,27,36,21,35,34,13);(2,4)(5,7,8,19,10,21,25,20,12,23,26,34,11,22,9,27)(6,13,14,18,16,29,24,33,17,30,15,31,28,36,32,35)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=f,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=ef,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93922, #93923, #93926, … |
| maximal normal subgroup of | #93922, #93923, #93926, … |
| maximal quotient of | #92913, #92914, #92915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 149);