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, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(25,26,30,28)(27,33,35,31)(29,36,38,32)(34,39,37,40);(11,19)(13,23)(15,24)(17,20)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,28,30,26)(27,31,35,33)(29,36,38,32)(34,39,37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=f,e^2=g,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×#46955, #7700737, #7700776, … |
| maximal normal subgroup of | C2×#46955, #7700737, #7700776, … |
| maximal quotient of | #576989, #577005, #577122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39943);