Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,7)(3,5)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,37)(39,40);(1,2,6,7)(3,4,5,8)(10,14)(11,17)(16,21)(18,23)(26,31)(28,34)(32,35)(36,40);(1,3,6,5)(2,8,7,4)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,30,38)(26,32,31,35)(28,36,34,40)(29,37,33,39);(1,4,6,8)(2,3,7,5)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,30,34)(26,33,31,29)(27,36,38,40)(32,39,35,37);(1,5,6,3)(2,8,7,4)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,33)(26,34,31,28)(27,37,38,39)(32,40,35,36)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52853, #8092689, #8092704, … |
| maximal normal subgroup of | C2×#52853, #8092689, #8092704, … |
| maximal quotient of | #582322, #582526, #582552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45841);