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 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32)(35,41)(37,45)(39,42)(40,44);(3,8)(5,7)(11,15)(13,16)(18,19)(20,27)(21,29)(30,31)(33,34,38,36)(35,42,41,39)(37,44,45,40)(43,46,47,48);(9,11,12,15)(10,13,14,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,3)(2,5)(4,8)(6,7)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36,38,34)(35,39,41,42)(37,44,45,40)(43,46,47,48);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=h,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, 6075120);