Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,31)(21,25)(22,26)(24,33)(27,45)(28,43)(29,39)(30,38)(32,48)(34,44)(35,42)(36,41)(37,46)(40,47);(1,2)(3,7)(4,8)(5,6)(17,19,21,29)(18,23,25,39)(20,32,33,47)(22,28,35,30)(24,40,31,48)(26,43,42,38)(27,36,46,34)(37,44,45,41);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,21,33)(18,24,25,31)(19,27,29,46)(22,36,35,34)(23,37,39,45)(26,44,42,41)(28,32,30,47)(38,48,43,40);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(20,33)(24,31)(27,46)(32,47)(34,36)(37,45)(40,48)(41,44);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(18,25)(19,28)(20,33)(22,35)(23,38)(27,47)(29,30)(32,46)(37,40)(39,43)(41,44)(45,48)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=gi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6646831);