Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#106 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15,17,31)(8,19)(9,12)(10,24)(11,16,26,32)(13,22)(14,23)(18,33,25,29)(20,27,35,30)(21,34)(28,36);(5,7,10,25,9,17,23,18)(6,11,14,30,12,26,24,27)(8,20,22,16,21,35,36,32)(13,29,19,15,28,33,34,31);(1,2,3,4)(6,12)(11,26)(13,28)(14,24)(15,31)(19,34)(27,30)(29,33)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=eh,d^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #145026, #145027, #145028, … |
| maximal normal subgroup of | #145026, #145027, #145028, … |
| maximal quotient of | #140389, #140404, #140422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6172);