Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | |
1368 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)(19,24)(20,25)(21,23)(22,26)(27,39)(28,38)(29,34)(30,44)(31,35)(32,36)(33,40)(37,42)(41,48)(43,46)(45,47);(1,2,5,6)(3,7,4,8)(18,23)(19,27)(20,30)(22,32)(24,37)(25,40)(29,42)(31,33)(34,39)(35,44)(38,46)(41,45);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,19,22,27,21,29,32,42)(18,24,26,39,23,34,36,37)(20,28,33,41,31,43,30,45)(25,38,40,48,35,46,44,47);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(18,23)(20,31)(24,34)(26,36)(28,43)(30,33)(37,39)(41,45);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,20,32,30,21,31,22,33)(18,25,36,44,23,35,26,40)(19,28,42,45,29,43,27,41)(24,38,37,47,34,46,39,48)Polycyclic presentation:
2^9|b^2=fi,c^2=fhi,d^2=g,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751527);