Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353473 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #63064 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2559 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,24,29)(19,30,33,35)(20,32,25,39)(21,26,36,41)(22,28,23,27)(31,44,45,47)(34,46,40,48)(37,43,38,42);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,25)(20,29)(21,31)(22,39)(23,32)(24,33)(26,40)(27,35)(28,30)(34,41)(36,45)(37,48)(38,46)(42,47)(43,44);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,24,25)(18,19,29,33)(21,34,36,40)(22,35,23,30)(26,31,41,45)(27,32,28,39)(37,47,38,44)(42,46,43,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21)(18,26)(19,31)(20,34)(22,37)(23,38)(24,36)(25,40)(27,42)(28,43)(29,41)(30,44)(32,46)(33,45)(35,47)(39,48);(21,36)(26,41)(31,45)(34,40)(37,38)(42,43)(44,47)(46,48)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260701);