Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18,20,22)(19,21,23,24);(2,5)(6,8)(10,14)(11,16)(17,18)(19,21)(20,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14);(1,3)(2,6)(4,7)(5,8)(9,12,13,15)(10,16,14,11)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=fg,c^2=g,e^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 | #161589, #161873, #162367, … |
| maximal normal subgroup of | #161589, #161873, #162367, … |
| maximal quotient of | #516464, #516474, #516640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28005);