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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
890 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)(3,8,7,5)(9,10,13,16)(11,12,15,14)(17,18,20,22)(19,21,23,24);(2,4)(3,7)(11,15)(12,14)(17,18)(19,21)(20,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,14,16,12)(17,18)(19,21)(20,22)(23,24);(1,4,6,2)(3,8,7,5)(9,12,13,14)(10,15,16,11);(1,5,6,8)(2,7,4,3)(9,10)(11,14)(12,15)(13,16)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=g,c^2=fg,d^2=h,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 | #97652, #161591, #161875, … |
| maximal normal subgroup of | #97652, #161591, #161875, … |
| maximal quotient of | #516466, #516477, #516642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28164);