Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10524 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,11)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(17,18,23,27)(19,28,32,43)(20,33,36,25)(21,38,22,26)(24,31,40,30)(29,44,46,41)(34,39,45,48)(35,42,37,47);(1,3)(2,7)(4,13)(5,14)(6,12)(8,16)(9,11)(10,15)(17,19,23,32)(18,24,27,40)(20,34,36,45)(21,31,22,30)(25,41,33,44)(26,43,38,28)(29,35,46,37)(39,47,48,42);(2,8)(3,12)(5,6)(9,10)(13,14)(15,16)(17,20,21,35,23,36,22,37)(18,25,26,42,27,33,38,47)(19,29,30,45,32,46,31,34)(24,39,28,41,40,48,43,44)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=dh,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,f^b=fgh,e^c=egh,f^c=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94679, #151253, C2×#10529, … |
| maximal normal subgroup of | #94679, #151253, C2×#10529, … |
| maximal quotient of | #104882, #104887, #104896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3517);