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 | |
716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,12)(11,15,17,19)(13,20,21,16)(18,24,23,22)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,3,5,8)(2,6,7,4)(10,12)(11,17)(13,21)(22,24)(26,31)(28,32)(34,37)(36,39);(2,7)(4,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,4,5,6)(2,3,7,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,16,12,20)(11,18,17,23)(15,22,19,24)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fgh,d^2=f,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×#56998, #8099664, #8100315, … |
| maximal normal subgroup of | C2×#56998, #8099664, #8100315, … |
| maximal quotient of | #586277, #586541, #587173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49986);