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 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,8,4)(9,10,12,14)(11,13,16,15)(21,33)(26,35)(29,40)(31,44)(37,45)(38,43)(42,47)(46,48);(9,10)(11,13)(12,14)(15,16)(18,23)(19,27)(25,30)(26,35)(28,39)(29,40)(38,43)(42,47);(1,3,5,8)(2,4,6,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,27,34)(20,30,32,25)(21,26,33,35)(28,41,39,36)(29,37,40,45)(31,43,44,38)(42,48,47,46);(1,4)(2,3)(5,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,28)(21,29)(22,27)(23,34)(25,36)(26,37)(30,41)(31,42)(32,39)(33,40)(35,45)(38,46)(43,48)(44,47);(17,20,22,32)(18,25,23,30)(19,28,27,39)(21,31,33,44)(24,36,34,41)(26,38,35,43)(29,42,40,47)(37,46,45,48);(17,21)(18,26)(19,29)(20,31)(22,33)(23,35)(24,37)(25,38)(27,40)(28,42)(30,43)(32,44)(34,45)(36,46)(39,47)(41,48)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=g,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581375);