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) = C26×C4 |
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 | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,15)(11,16,12,14)(18,21)(22,24);(2,6)(3,7)(9,11)(10,14)(12,13)(15,16)(19,23)(22,24);(11,12)(14,16)(17,18,20,21)(19,22,23,24);(1,3,5,7)(2,4,6,8)(9,12,13,11)(10,14,15,16)(17,18)(19,22)(20,21)(23,24);(1,4)(2,7)(3,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=f,e^2=g,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×#54486, #8100764, #8101184, … |
| maximal normal subgroup of | C2×#54486, #8100764, #8101184, … |
| maximal quotient of | #585634, #586174, #586998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47474);