Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,10,6,8,4,9)(3,11,14,7,13,16,12,15)(17,18,22,26,23,28,21,27)(19,29,32,39,33,42,31,24)(20,34,36,25,37,40,35,41)(30,38,45,47,46,48,44,43);(1,3)(2,7)(4,14)(5,12)(6,13)(8,15)(9,16)(10,11)(17,19)(18,24,28,39)(20,30)(21,31)(22,32)(23,33)(25,38,41,48)(26,29,27,42)(34,43,40,47)(35,44)(36,45)(37,46);(2,8)(4,5)(7,15)(12,14)(17,20,23,37)(18,25,28,41)(19,30,33,46)(21,36,22,35)(24,38,39,48)(26,34,27,40)(29,43,42,47)(31,45,32,44)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153306, C2×#9426, #200838, … |
| maximal normal subgroup of | #153306, C2×#9426, #200838, … |
| maximal quotient of | #100092, #100102, #100772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2414);