Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,12,15,20)(13,16,22,24)(18,19,21,23)(25,26,30,35)(27,32,38,31)(28,33,34,29)(36,37,39,40);(9,11,14,15)(10,12,17,20)(13,21,22,18)(16,23,24,19)(25,27,30,38)(26,31,35,32)(28,39,34,36)(29,40,33,37);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,18)(14,15)(16,19)(17,20)(21,22)(23,24)(25,27)(26,32)(28,39)(29,37)(30,38)(31,35)(33,40)(34,36);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,19)(14,20)(16,21)(18,24)(22,23)(25,28,30,34)(26,33,35,29)(27,36,38,39)(31,40,32,37);(9,13)(10,16)(11,18)(12,19)(14,22)(15,21)(17,24)(20,23)(25,29,30,33)(26,34,35,28)(27,37,38,40)(31,39,32,36)Polycyclic presentation:
2^8|a^2=gh,b^2=h,c^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47311, #7701421, #7701443, … |
| maximal normal subgroup of | C2×#47311, #7701421, #7701443, … |
| maximal quotient of | #576837, #576870, #577069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40299);