Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1510 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)(20,33,36,25)(21,26,23,28)(30,39,46,45)(31,44,38,40)(34,41,47,48)(35,43,37,42);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,27,29)(20,34,36,47)(21,31,23,38)(25,41,33,48)(26,40,28,44)(30,35,46,37)(39,42,45,43);(10,14)(13,15)(17,20,22,36)(18,25,27,33)(19,30,32,46)(21,37,23,35)(24,39,29,45)(26,43,28,42)(31,47,38,34)(40,48,44,41);(2,5)(6,8)(11,16)(13,15)(17,19)(18,24)(20,30)(21,38)(22,32)(23,31)(25,39)(26,44)(27,29)(28,40)(33,45)(34,37)(35,47)(36,46)(41,43)(42,48);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,28)(19,31,32,38)(20,35,36,37)(24,40,29,44)(25,42,33,43)(30,34,46,47)(39,41,45,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419505);