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 | |
705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26)(27,33)(28,36)(29,35)(30,31)(32,37)(34,39)(38,40);(1,2,5,7)(3,4,8,6)(11,18)(14,17)(19,24)(21,23)(26,31)(28,37)(29,33)(34,38);(1,3,5,8)(2,6,7,4)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,27)(26,29)(28,38)(30,35)(31,33)(32,40)(34,37)(36,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,34)(29,39)(30,37)(31,36)(33,40)(35,38);(1,4)(2,3)(5,6)(7,8)(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29)(26,27)(28,39)(30,33)(31,35)(32,34)(36,38)(37,40)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,c^2=fgh,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×#57462, #8100556, #8101197, … |
| maximal normal subgroup of | C2×#57462, #8100556, #8101197, … |
| maximal quotient of | #586216, #586550, #587112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50450);