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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26,30,33)(27,28,31,36)(29,32,38,40)(34,35,37,39);(1,3)(2,6)(4,7)(5,8)(13,21)(16,23)(18,24)(20,22)(25,27,30,31)(26,28,33,36)(29,37,38,34)(32,39,40,35);(1,4,5,6)(2,3,7,8)(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,27)(26,28)(29,34)(30,31)(32,35)(33,36)(37,38)(39,40);(1,4)(2,3)(5,6)(7,8)(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,37)(34,40)(38,39);(1,2)(3,4)(5,7)(6,8)(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,40)(36,39)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=g,d^2=fg,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×#56233, #8101675, #8101953, … |
| maximal normal subgroup of | C2×#56233, #8101675, #8101953, … |
| maximal quotient of | #585630, #586190, #586713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49221);