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:
(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(18,23)(19,27)(21,34)(25,36)(29,31)(32,45)(38,42)(47,48);(1,2,4,6)(3,5,7,8)(10,14)(11,15)(18,23)(24,28)(25,36)(26,35)(37,39)(38,42)(40,46)(47,48);(1,3)(2,5)(4,7)(6,8)(9,11,13,15)(10,12,14,16)(17,18)(19,28)(20,25)(21,35)(22,23)(24,27)(26,34)(29,39)(30,38)(31,37)(32,46)(33,36)(40,45)(41,42)(43,47)(44,48);(9,12)(10,15)(11,14)(13,16)(17,19,22,27)(18,24,23,28)(20,31,33,29)(21,30,34,41)(25,39,36,37)(26,38,35,42)(32,44,45,43)(40,48,46,47);(1,2)(3,5)(4,6)(7,8)(17,20,22,33)(18,25,23,36)(19,29,27,31)(21,32,34,45)(24,37,28,39)(26,40,35,46)(30,43,41,44)(38,47,42,48);(1,3)(2,5)(4,7)(6,8)(17,21)(18,26)(19,30)(20,32)(22,34)(23,35)(24,38)(25,40)(27,41)(28,42)(29,43)(31,44)(33,45)(36,46)(37,47)(39,48)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839731);