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 | |
1125 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)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(18,21)(19,23);(2,5)(6,7)(9,11,13,16)(10,14,12,15)(17,18,20,21)(19,24,23,22);(11,16)(14,15)(17,18,20,21)(19,22,23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,14,16,15)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|b^2=gh,c^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×#54933, #8098904, #8099301, … |
| maximal normal subgroup of | C2×#54933, #8098904, #8099301, … |
| maximal quotient of | #585223, #585639, #585875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47921);