Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1495 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)(17,18,22,27)(19,29,32,24)(20,25,35,42)(21,26,23,28)(30,45,47,38)(31,43,36,39)(33,48,46,40)(34,41,37,44);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,27,29)(20,33,35,46)(21,31,23,36)(25,40,42,48)(26,39,28,43)(30,37,47,34)(38,44,45,41);(10,14)(13,15)(17,20,22,35)(18,25,27,42)(19,30,32,47)(21,34,23,37)(24,38,29,45)(26,41,28,44)(31,46,36,33)(39,48,43,40);(2,5)(6,8)(11,16)(13,15)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,38)(26,43)(27,29)(28,39)(33,34)(35,47)(37,46)(40,41)(42,45)(44,48);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,43)(25,41,42,44)(30,46,47,33)(38,48,45,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419184);