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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,26,30,28)(27,33,35,31)(29,36,37,32)(34,38,40,39);(1,2,5,4)(3,6,7,8)(10,14)(11,17)(12,21)(22,24)(26,28)(31,33)(32,36)(38,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,5,2)(3,8,7,6)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,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×#47354, #7701324, #7701450, … |
| maximal normal subgroup of | C2×#47354, #7701324, #7701450, … |
| maximal quotient of | #576986, #576987, #577117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40342);