Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6369207 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9)(3,7,5,13)(4,11,12,15)(8,14,16,10)(17,18,23,28)(19,29,33,39)(20,25,36,43)(21,27,22,26)(24,31,40,32)(30,45,46,41)(34,48,47,38)(35,44,37,42);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,23,33)(18,24,28,40)(20,34,36,47)(21,32,22,31)(25,41,43,45)(26,29,27,39)(30,37,46,35)(38,44,48,42);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,42)(27,44)(28,43)(29,45)(31,34)(32,47)(33,46)(39,41)(40,48);(2,9)(7,13)(8,16)(11,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,42,43,44)(30,34,46,47)(38,41,48,45)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=i,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276892);