Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357891 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3498 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1867 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18)(19,28)(20,26)(21,27)(22,36)(23,25)(24,30)(29,43)(31,39)(32,41)(33,40)(34,37)(35,42)(38,45)(44,47)(46,48);(2,6)(7,14)(8,13)(9,15)(17,19,21,34,23,31,22,30)(18,24,36,39,25,37,27,28)(20,29,35,44,33,46,32,45)(26,38,41,48,40,47,42,43);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(18,25)(24,37)(26,40)(27,36)(28,39)(38,47)(41,42)(43,48);(3,10)(7,14)(9,15)(11,16)(17,20)(18,26)(19,29)(21,35)(22,32)(23,33)(24,38)(25,40)(27,42)(28,43)(30,45)(31,46)(34,44)(36,41)(37,47)(39,48);(17,21,23,22)(18,27,25,36)(19,30,31,34)(20,32,33,35)(24,39,37,28)(26,41,40,42)(29,44,46,45)(38,43,47,48)Polycyclic presentation:
2^9|b^2=fhi,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265117);