Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907977 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1720 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)(21,26)(22,36)(23,27)(25,34)(28,42)(29,38)(30,39)(32,41)(33,45)(35,40)(37,43)(44,47)(46,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,30)(18,24,27,39)(20,28,33,44)(21,35,22,29)(25,37,41,48)(26,40,36,38)(31,42,45,47)(32,46,34,43);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,21,32,23,33,22,34)(18,25,36,45,27,41,26,31)(19,28,35,46,30,44,29,43)(24,37,38,47,39,48,40,42);(2,6)(5,7)(11,16)(13,15)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,32,33,34)(24,38,39,40)(25,31,41,45)(28,43,44,46)(37,47,48,42);(10,14)(13,15)(19,30)(20,33)(24,39)(25,41)(29,35)(31,45)(32,34)(38,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815779);