Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,22,27)(19,28,23,24)(20,31,33,42)(21,34,36,26)(25,37,40,29)(30,38,35,41)(32,45,47,44)(39,43,48,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,29,33,37)(21,35,36,30)(25,31,40,42)(26,41,34,38)(32,46,47,43)(39,45,48,44);(3,8)(5,7)(10,13)(14,16)(17,20,22,33)(18,25,27,40)(19,29,23,37)(21,32,36,47)(24,31,28,42)(26,39,34,48)(30,43,35,46)(38,44,41,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,32)(22,36)(23,35)(24,38)(25,39)(27,34)(28,41)(29,43)(31,44)(33,47)(37,46)(40,48)(42,45);(10,13)(14,16)(21,36)(26,34)(30,35)(32,47)(38,41)(39,48)(43,46)(44,45)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678702);