Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,13)(11,12)(18,19)(20,24)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(1,3,5,8)(2,6,7,4)(9,10,14,13)(11,16,12,15)(17,18,21,19)(20,22,24,23)(26,27)(28,35)(32,34)(36,40);(2,7)(3,8)(9,11,14,12)(10,15,13,16)(17,19,21,18)(20,22,24,23)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,4,5,6)(2,8,7,3)(9,12,14,11)(10,15,13,16)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,2)(3,6)(4,8)(5,7)(9,13,14,10)(11,15,12,16)(17,20,21,24)(18,22,19,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=fhi,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2456190);