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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(9,10)(11,16)(12,15)(13,14)(18,22)(23,25)(24,29)(30,31)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(1,2,4,6)(3,5,7,8)(9,11,14,12)(10,15,13,16)(20,28)(24,29)(26,32)(30,31)(34,36)(39,43)(40,44)(47,48);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,15,13,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,39)(36,43)(37,45)(38,42)(40,48)(41,46)(44,47);(9,10,14,13)(11,15,12,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,3)(2,5)(4,7)(6,8)(9,13,14,10)(11,16,12,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,46)(34,40,36,44)(35,41,42,45)(39,47,43,48)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=gi,d^2=fg,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, 6189454);