Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,26)(21,27)(22,28)(23,24)(29,32)(30,41)(31,35)(33,42)(34,39)(36,37)(38,40)(43,45)(44,47)(46,48);(3,7)(6,8)(9,10,12,14)(11,13,16,15)(18,24)(21,25)(26,39)(28,40)(32,35)(36,41)(42,48)(45,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,25)(20,31)(21,24)(22,37)(23,27)(26,35)(28,36)(29,34)(30,38)(32,39)(33,43)(40,41)(42,45)(44,46)(47,48);(9,11,12,16)(10,13,14,15)(17,20,23,34)(18,26,24,39)(19,29,27,31)(21,35,25,32)(22,33,38,46)(28,42,40,48)(30,43,37,44)(36,47,41,45);(1,3)(2,6)(4,7)(5,8)(17,21,23,25)(18,27,24,19)(20,32,34,35)(22,36,38,41)(26,29,39,31)(28,37,40,30)(33,45,46,47)(42,43,48,44);(1,2)(3,6)(4,5)(7,8)(17,22)(18,28)(19,30)(20,33)(21,36)(23,38)(24,40)(25,41)(26,42)(27,37)(29,43)(31,44)(32,45)(34,46)(35,47)(39,48)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961048);