Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1457 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)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,33)(21,26)(22,27)(23,28)(25,36)(29,45)(30,39)(31,43)(32,40)(34,48)(35,44)(37,42)(38,47)(41,46);(1,3)(2,5)(4,8)(6,7)(17,19,22,32)(18,24,27,40)(20,34,36,46)(21,30,23,31)(25,41,33,48)(26,39,28,43)(29,37,47,35)(38,44,45,42);(10,14)(15,16)(17,20,22,36)(18,25,27,33)(19,29,32,47)(21,35,23,37)(24,38,40,45)(26,42,28,44)(30,46,31,34)(39,48,43,41);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,23)(18,26,27,28)(19,30,32,31)(20,35,36,37)(24,39,40,43)(25,42,33,44)(29,46,47,34)(38,48,45,41);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,27)(19,31)(20,36)(21,23)(24,39)(29,46)(30,32)(34,47)(38,41)(40,43)(42,44)(45,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=g,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6636877);