Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #106 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | C4×#106 |
| prime radical | C23 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,7,14,20,27,8,13,5,11,19,26,4,10)(3,15,16,25,17,28,21,24,18,31,32,30,23,29,22,9);(1,3)(2,9,13,24)(4,16)(5,21)(6,22)(7,23)(8,18)(10,15,27,31)(11,28,12,29)(14,30,26,25)(17,19)(20,32);(1,4,19,5,8,20,7,6)(2,10,26,11,13,27,14,12)(3,16,17,21,18,32,23,22)(9,15,25,28,24,31,30,29)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=g,e^2=g,g^2=h,b^a=bd,f^a=fh,d^b=dg,e^b=egh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94417, #94418, #129137, … |
| maximal normal subgroup of | #94417, #94418, #129137, … |
| maximal quotient of | #129088, #129104, #129126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5656);