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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,16)(11,12,13,14)(18,22)(19,25)(24,29)(26,31);(2,5)(6,7)(9,11,15,13)(10,12,16,14)(18,22)(23,28)(24,29)(30,32);(9,12,15,14)(10,11,16,13)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,12,16,14)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,3,4,8)(2,6,5,7)(9,14,15,12)(10,11,16,13)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,c^2=g,d^2=gh,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×#58581, #8104193, #8104425, … |
| maximal normal subgroup of | C2×#58581, #8104193, #8104425, … |
| maximal quotient of | #585876, #586159, #586863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51569);