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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1466 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,40)(30,31)(32,42)(33,43)(34,39)(35,37)(36,44)(38,41)(45,47)(46,48);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,30)(22,38)(23,32)(25,43)(26,31)(27,42)(28,41)(29,35)(34,48)(36,45)(37,40)(39,46)(44,47);(10,14)(13,16)(17,20,23,36)(18,25,27,44)(19,29,32,46)(21,37,28,34)(22,39,26,35)(24,40,42,48)(30,47,41,43)(31,45,38,33);(1,4,6,5)(2,7,3,8)(17,21,23,28)(18,26,27,22)(19,30,32,41)(20,34,36,37)(24,31,42,38)(25,39,44,35)(29,43,46,47)(33,48,45,40);(1,5,6,4)(2,7,3,8)(18,27)(19,31)(22,26)(24,41)(25,44)(29,45)(30,42)(32,38)(33,46)(35,39)(40,43)(47,48)Polycyclic presentation:
2^9|b^2=fg,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,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, 6588956);