Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
974 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)(10,14)(11,16)(18,21)(22,24);(2,5)(6,8)(9,10,13,14)(11,15,16,12)(19,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12)(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,7)(5,8)(9,12,13,15)(10,16,14,11);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53165, #8105656, #8105700, … |
| maximal normal subgroup of | C2×#53165, #8105656, #8105700, … |
| maximal quotient of | #606712, #606715, #606718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46153);