Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 |
| prime radical | #2671 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
820 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,13,12,16)(18,21)(19,23);(2,5)(6,7)(10,15)(11,12)(17,18,20,21)(19,24,23,22);(9,11,14,12)(10,16,15,13)(17,18,20,21)(19,22,23,24);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,15,16)(17,18)(19,22)(20,21)(23,24);(1,3,4,8)(2,6,5,7)(9,13)(10,11)(12,15)(14,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58501, #8104228, #8104433, … |
| maximal normal subgroup of | C2×#58501, #8104228, #8104433, … |
| maximal quotient of | #585887, #586165, #586604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51489);