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 | #35687 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2226 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,27,29,23)(20,31,34,24)(22,26,35,42)(28,43,36,38)(30,39,33,41)(32,44,37,40)(45,47,46,48);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(19,28)(20,32)(22,35)(23,38)(24,40)(26,42)(27,43)(29,36)(31,44)(34,37)(45,46)(47,48);(10,14)(13,16)(17,19)(18,23)(20,33)(21,29)(22,36)(24,41)(25,27)(26,43)(28,35)(30,34)(31,39)(32,46)(37,45)(38,42)(40,48)(44,47);(2,6)(5,7)(19,29)(23,27)(28,36)(30,33)(38,43)(39,41)(45,46)(47,48);(3,8)(5,7)(17,20,21,34)(18,24,25,31)(19,30,29,33)(22,37,35,32)(23,39,27,41)(26,44,42,40)(28,45,36,46)(38,47,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,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, 6585497);