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 | |
1292 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)(11,15)(12,16)(13,14)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,32,30,31)(35,41)(38,42)(43,47)(45,48);(3,8)(5,7)(9,11)(10,15)(12,14)(13,16)(18,20)(21,29)(23,25)(26,30)(33,34,37,40)(35,42,41,38)(36,39,44,46)(43,48,47,45);(9,10,14,13)(11,15,12,16)(17,19)(18,23)(20,25)(21,30)(22,27)(24,32)(26,29)(28,31)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(1,3)(2,5)(4,8)(6,7)(9,12,14,11)(10,15,13,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,32,30,31)(33,35)(34,38)(36,43)(37,41)(39,45)(40,42)(44,47)(46,48);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,15,12,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,31)(25,32)(27,30)(33,36,37,44)(34,39,40,46)(35,43,41,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=fg,d^2=g,e^2=fgh,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, 6087353);