Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,13,15)(11,12,16,14)(18,21)(19,23)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(1,3,5,8)(2,4,6,7)(9,10)(11,12)(13,15)(14,16)(19,23)(22,24)(26,31)(28,35)(33,38)(36,40);(3,8)(4,7)(9,11,13,16)(10,14,15,12)(17,18,20,21)(19,22,23,24)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,38)(37,40);(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,36,35,40)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371165);