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 | |
714 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,14)(13,20)(15,19)(16,21)(18,22)(23,24)(25,26,30,34)(27,28,32,31)(29,37,39,33)(35,40,38,36);(4,8)(6,7)(10,12)(13,21)(15,17)(18,24)(25,27,30,32)(26,31,34,28)(29,38,39,35)(33,36,37,40);(2,3)(6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,40)(38,39);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28)(26,32)(27,34)(29,36)(30,31)(33,38)(35,37)(39,40);(1,4,5,8)(2,6,3,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,39)(26,33,34,37)(27,35,32,38)(28,36,31,40)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=fh,d^2=g,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×#57860, #8101568, #8101957, … |
| maximal normal subgroup of | C2×#57860, #8101568, #8101957, … |
| maximal quotient of | #585678, #586340, #586744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50848);