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 | #2828 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(26,30)(27,33)(28,36)(38,40);(2,6)(3,7)(10,14)(11,17)(12,21)(22,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,3)(2,4)(5,7)(6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=h,e^2=g,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×#56861, #8100133, #8100312, … |
| maximal normal subgroup of | C2×#56861, #8100133, #8100312, … |
| maximal quotient of | #586668, #587031, #587181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49849);