Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1605 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)(11,14)(12,13)(15,16)(17,18)(19,24)(20,26)(21,34)(22,25)(23,28)(27,36)(29,43)(30,38)(31,46)(32,40)(33,44)(35,48)(37,41)(39,47)(42,45);(1,3)(2,5)(4,8)(6,7)(17,19,22,32)(18,24,25,40)(20,30,23,29)(21,35,36,45)(26,38,28,43)(27,42,34,48)(31,37,47,33)(39,44,46,41);(10,13)(14,16)(18,25)(19,29)(20,23)(21,36)(24,38)(30,32)(31,45)(35,47)(39,42)(40,43)(41,44)(46,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,23)(18,26,25,28)(19,30,32,29)(21,33,36,37)(24,38,40,43)(27,41,34,44)(31,45,47,35)(39,48,46,42);(3,8)(5,7)(17,21,22,36)(18,27,25,34)(19,31,32,47)(20,33,23,37)(24,39,40,46)(26,41,28,44)(29,35,30,45)(38,48,43,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6602921);