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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,7,8)(3,4,6,5)(9,10,13,16)(11,14,12,15)(17,18,21,20)(19,22,23,24)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,40)(38,39);(1,3,7,6)(2,5,8,4)(19,23)(22,24)(26,31)(28,35)(33,39)(36,40);(1,4,7,5)(2,3,8,6)(9,11,13,12)(10,14,16,15)(17,19,21,23)(18,22,20,24)(25,27,30,32)(26,28,31,35)(29,37,38,34)(33,40,39,36);(1,5,7,4)(2,3,8,6)(9,10)(11,14)(12,15)(13,16)(17,18,21,20)(19,22,23,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,37)(34,39)(38,40);(1,6,7,3)(2,4,8,5)(9,12,13,11)(10,15,16,14)(17,20,21,18)(19,24,23,22)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=hi,d^2=h,e^2=fghi,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, 2456202);