Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 131072 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10)(11,16)(12,13)(14,15)(17,18,20,22)(19,21,23,24);(2,5)(6,8)(10,15)(11,12)(17,18)(19,21)(20,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,16,15,13)(17,18)(19,21)(20,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,15,16)(17,19,20,23)(18,21,22,24);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,15)(14,16)Polycyclic presentation:
2^8|a^2=g,c^2=fg,d^2=f,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 | #97141, #97143, #207932, … |
| maximal normal subgroup of | #97141, #97143, #207932, … |
| maximal quotient of | #516470, #516643, #516722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28002);