Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60697 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9)(3,10,12,7)(4,13,5,8)(11,15,14,16)(17,18,23,28)(19,30,34,24)(20,36,38,44)(21,29,40,26)(22,27,37,47)(25,33,45,39)(31,43,41,46)(32,42,35,48);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19,23,34)(18,24,28,30)(20,37,38,22)(21,32,40,35)(25,43,45,46)(26,42,29,48)(27,44,47,36)(31,39,41,33);(17,20,21,33,23,38,40,39)(18,25,29,36,28,45,26,44)(19,31,35,37,34,41,32,22)(24,27,42,43,30,47,48,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,40)(18,26,28,29)(19,32,34,35)(20,33,38,39)(22,41,37,31)(24,42,30,48)(25,44,45,36)(27,43,47,46);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,22,21,31,23,37,40,41)(18,27,29,43,28,47,26,46)(19,33,35,38,34,39,32,20)(24,25,42,36,30,45,48,44)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=f,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719191);