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 | |
695 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)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,31,29)(27,34,28,32)(30,37,38,33)(35,39,36,40);(10,14)(11,18)(12,21)(22,23)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3,4,7)(2,6,5,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=fg,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×#47298, #7701402, #7701454, … |
| maximal normal subgroup of | C2×#47298, #7701402, #7701454, … |
| maximal quotient of | #576806, #576880, #577136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40286);