Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1223 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)(3,7)(4,8)(5,6)(9,10,13,12)(11,14,16,15)(17,18)(19,24)(20,33)(21,38)(22,27)(23,29)(25,36)(26,32)(28,35)(30,41)(31,40)(34,39)(37,43)(42,46)(44,45)(47,48);(2,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,23,32)(18,24,29,26)(20,34,37,46)(21,22,31,35)(25,41,44,48)(27,40,28,38)(30,45,47,36)(33,39,43,42);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,15,16,14)(17,20,35,45,23,37,22,36)(18,25,27,43,29,44,28,33)(19,30,31,46,32,47,21,34)(24,39,38,48,26,42,40,41);(11,16)(14,15)(17,21,22,32,23,31,35,19)(18,26,28,38,29,24,27,40)(20,34,36,47,37,46,45,30)(25,42,33,48,44,39,43,41);(1,4,5,3)(2,7,6,8)(17,22,23,35)(18,27,29,28)(19,31,32,21)(20,36,37,45)(24,40,26,38)(25,43,44,33)(30,46,47,34)(39,41,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=fhi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7512028);