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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18,24,30)(19,25,33,41)(20,26,38,46)(21,27,23,29)(22,28,32,40)(31,39,47,48)(34,42,36,44)(35,43,37,45);(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,34)(21,32)(22,23)(24,33)(26,42)(27,40)(28,29)(30,41)(31,37)(35,47)(36,38)(39,45)(43,48)(44,46);(2,5)(6,8)(10,14)(11,16)(17,20,23,37,24,38,21,35)(18,26,29,45,30,46,27,43)(19,31,32,34,33,47,22,36)(25,39,40,42,41,48,28,44);(10,14)(13,15)(17,21,24,23)(18,27,30,29)(19,32,33,22)(20,35,38,37)(25,40,41,28)(26,43,46,45)(31,34,47,36)(39,42,48,44);(1,3,4,7)(2,6,5,8)(17,22,21,33,24,32,23,19)(18,28,27,41,30,40,29,25)(20,36,35,47,38,34,37,31)(26,44,43,48,46,42,45,39)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fh,d^2=i,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, 6740164);