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 | |
1585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,30)(20,34)(21,29)(22,27)(23,28)(24,32)(25,38)(26,39)(31,40)(33,44)(35,41)(36,42)(37,47)(43,48)(45,46);(1,2,4,5)(3,6,7,8)(17,19)(18,24)(20,35)(21,22)(23,32)(25,41)(26,27)(28,30)(29,44)(31,36)(33,39)(34,47)(37,38)(40,43)(42,45)(46,48);(9,11,13,12)(10,14,15,16)(17,20,21,36,23,38,39,48)(18,25,29,43,28,34,26,42)(19,31,33,35,32,46,22,37)(24,40,27,41,30,45,44,47);(1,3,4,7)(2,6,5,8)(17,21,23,39)(18,26,28,29)(19,22,32,33)(20,36,38,48)(24,27,30,44)(25,42,34,43)(31,37,46,35)(40,41,45,47);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,22,21,19,23,33,39,32)(18,27,29,30,28,44,26,24)(20,37,36,31,38,35,48,46)(25,41,43,45,34,47,42,40)Polycyclic presentation:
2^9|b^2=fi,c^2=ghi,d^2=f,e^2=ghi,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, 6744158);