Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,5,8,7)(11,15)(13,16)(17,18,20,22)(19,23,24,21);(2,4)(3,8)(9,10,12,14)(11,13,15,16)(17,18)(19,21)(20,22)(23,24);(1,3,6,8)(2,7,4,5)(9,11,12,15)(10,13,14,16)(17,19,20,24)(18,21,22,23);(1,4,6,2)(3,5,8,7);(1,5)(2,3)(4,8)(6,7)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=g,d^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #122008, #122028, #122076, … |
| maximal normal subgroup of | #122008, #122028, #122076, … |
| maximal quotient of | #516435, #516595, #516976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28169);