Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8790 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #382 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,8)(4,13)(5,15)(6,10)(7,11)(9,14)(12,16)(17,18)(19,27,30,23)(20,31,33,46)(21,26)(22,25)(24,32,37,47)(28,41)(29,40,44,36)(34,38)(35,42)(39,48)(43,45);(1,3,5,11,6,12,14,4)(2,7,9,8,10,13,15,16)(17,19,21,29,22,30,34,44)(18,23,26,36,25,27,38,40)(20,28,32,42,33,43,47,48)(24,35,31,45,37,39,46,41);(1,4,14,12,6,11,5,3)(2,8,15,7,10,16,9,13)(17,20,22,33)(18,24)(19,28,30,43)(21,32,34,47)(23,35)(25,37)(26,31)(27,39)(29,42,44,48)(36,45)(38,46)(40,41)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=dg,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fgh,f^c=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94579, #94581, #94582, … |
| maximal normal subgroup of | #94579, #94581, #94582, … |
| maximal quotient of | #99278, #99281, #99285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1780);