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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,7,8)(3,4,5,6)(10,14)(11,17)(16,21)(18,23)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,39,38,40);(1,3,7,5)(2,4,8,6)(10,14)(15,20)(16,21)(22,24)(26,30)(28,37)(31,33)(34,38);(1,4,7,6)(2,3,8,5)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,31)(28,38)(29,35)(30,33)(32,40)(34,37)(36,39);(1,5,7,3)(2,4,8,6)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,29,37)(26,32,30,36)(27,34,35,38)(31,39,33,40);(1,6,7,4)(2,3,8,5)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,37)(38,40)Polycyclic presentation:
2^8|a^2=gh,b^2=fgh,c^2=fgh,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×#57951, #8101488, #8101954, … |
| maximal normal subgroup of | C2×#57951, #8101488, #8101954, … |
| maximal quotient of | #586101, #586350, #586480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50939);