Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,24)(20,31)(21,34)(22,27)(23,28)(25,29)(26,36)(30,42)(32,45)(33,38)(35,39)(37,41)(40,46)(43,48)(44,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,28,27)(20,29,33,37)(21,30,36,35)(25,38,41,31)(26,39,34,42)(32,43,47,46)(40,45,48,44);(9,11,12,16)(10,13,14,15)(17,20,23,33)(18,25,28,41)(19,29,22,37)(21,32,36,47)(24,38,27,31)(26,40,34,48)(30,43,35,46)(39,45,42,44);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,30)(20,32)(22,35)(23,36)(24,39)(25,40)(27,42)(28,34)(29,43)(31,44)(33,47)(37,46)(38,45)(41,48);(2,6)(5,7)(10,14)(13,15)(17,19,23,22)(18,27,28,24)(20,29,33,37)(21,35,36,30)(25,31,41,38)(26,39,34,42)(32,46,47,43)(40,45,48,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7268597);