Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
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 | |
1085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,21,20,24);(2,5)(3,8)(9,11,13,16)(10,14,12,15)(18,23)(19,20);(2,5)(6,7)(11,16)(14,15)(17,19,22,20)(18,24,23,21);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,19)(18,21,23,24);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,14,16,15)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=fgh,d^2=fgh,e^2=gh,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.
| Fitting subgroup of | #58989⋊C3 |
| maximal subgroup of | #132845, #486871, #486999, … |
| maximal normal subgroup of | #132845, #486871, #486999, … |
| maximal quotient of | #585261, #587542, #590033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51977);