Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(18,20)(19,21);(1,2,4,6)(3,5,8,7)(10,12)(11,15)(17,18)(19,23)(20,22)(21,24);(9,11,13,15)(10,14,12,16)(17,19,22,21)(18,23,20,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,18)(19,23,21,24);(1,3,4,8)(2,5,6,7)(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^8|b^2=fgh,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#57299, #8100823, #8101181, … |
| maximal normal subgroup of | C2×#57299, #8100823, #8101181, … |
| maximal quotient of | #585653, #585989, #586200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50287);