Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1392 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,21,20)(19,22,24,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(3,8)(4,7)(9,11,12,15)(10,13,14,16)(17,19,21,24)(18,22,20,23)(26,30)(31,33)(32,38)(39,40);(2,6)(3,8)(9,11)(10,13)(12,15)(14,16)(17,20,21,18)(19,22,24,23)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,37)(38,39);(1,3,5,8)(2,7,6,4)(17,19)(18,23)(20,22)(21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,4,5,7)(2,8,6,3)(9,10)(11,13)(12,14)(15,16)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6143981);