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) = C25×C4 |
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 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,15,16,13)(18,20)(22,23)(26,28)(29,37)(31,35)(33,38);(4,8)(6,7)(11,16)(13,15)(18,20)(22,23)(25,26,30,28)(27,31,34,35)(29,32,37,36)(33,39,38,40);(2,3)(6,7)(17,18,21,20)(19,23,24,22)(25,27)(26,31)(28,35)(29,38)(30,34)(32,40)(33,37)(36,39);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,24)(18,22,20,23)(25,28,30,26)(27,31,34,35)(29,36,37,32)(33,39,38,40);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,23,24,22)(25,29)(26,32)(27,33)(28,36)(30,37)(31,39)(34,38)(35,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=h,d^2=fg,e^2=g,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, 6075054);