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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,15,16)(11,12,13,14)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(1,2,5,7)(3,4,8,6)(9,11,15,13)(10,12,16,14)(18,22)(23,29)(24,27)(30,31);(1,3,5,8)(2,6,7,4)(9,12,15,14)(10,11,16,13)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(1,3)(2,4)(5,8)(6,7)(9,13,15,11)(10,12,16,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32);(1,4)(2,3)(5,6)(7,8)(9,14,15,12)(10,11,16,13)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=fgh,d^2=g,e^2=g,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×#57571, #8100951, #8101185, … |
| maximal normal subgroup of | C2×#57571, #8100951, #8101185, … |
| maximal quotient of | #586134, #586527, #587263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50559);