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 | |
707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(2,5)(6,8)(10,11)(12,19)(13,21)(22,23)(26,28)(27,34)(32,36)(35,40);(1,2,4,5)(3,6,7,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=gh,c^2=fg,d^2=g,e^2=f,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×#54284, #8103020, #8103195, … |
| maximal normal subgroup of | C2×#54284, #8103020, #8103195, … |
| maximal quotient of | #585950, #586530, #586944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47272);