Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,8,4,7)(9,10,13,16)(11,12,15,14)(17,18,20,22)(19,23,24,21);(2,6)(7,8)(11,15)(12,14);(1,3,5,4)(2,7,6,8)(9,11,13,15)(10,14,16,12)(17,19,20,24)(18,21,22,23);(1,2)(3,7)(4,8)(5,6)(9,12,13,14)(10,15,16,11);(1,4,5,3)(2,8,6,7)(9,10)(11,14)(12,15)(13,16)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #98095, #167939, #167940, … |
| maximal normal subgroup of | #98095, #167939, #167940, … |
| maximal quotient of | #553800, #553806, #554046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30124);