Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,12,14)(11,13,15,16)(19,23)(21,24)(25,26)(27,28)(29,37)(30,31)(32,38)(33,36)(34,40)(35,39);(2,5)(6,8)(11,15)(13,16)(17,18,20,22)(19,21,23,24)(26,31)(27,33)(29,38)(35,40);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(25,27,30,33)(26,28,31,36)(29,34,38,39)(32,35,37,40);(9,11)(10,13)(12,15)(14,16)(17,19,20,23)(18,21,22,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4419674);