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 | #2733 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(10,14)(12,16)(19,23)(21,24);(1,2,5,6)(3,7,4,8)(10,14)(11,15)(17,18,20,22)(19,21,23,24);(1,3,5,4)(2,7,6,8)(9,10,13,14)(11,12,15,16)(17,19,20,23)(18,21,22,24);(1,2)(3,7)(4,8)(5,6)(9,11,13,15)(10,12,14,16);(1,4,5,3)(2,8,6,7)(9,12)(10,15)(11,14)(13,16)Polycyclic presentation:
2^8|b^2=gh,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 | #98087, #167863, #167864, … |
| maximal normal subgroup of | #98087, #167863, #167864, … |
| maximal quotient of | #553780, #553800, #554011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30187);