Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357957 |
| 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 | #62535 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(4,11)(6,12)(8,14)(10,15)(13,16)(17,18)(19,28)(20,26)(21,27)(22,36)(23,25)(24,31)(29,43)(30,37)(32,42)(33,40)(34,39)(35,41)(38,44)(45,48)(46,47);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,22,31,23,32,21,30)(18,24,27,28,25,37,36,42)(20,29,33,44,34,45,35,46)(26,38,41,43,39,47,40,48);(1,3,4,9,5,10,11,15)(2,6,8,12,7,13,14,16)(18,25)(24,37)(26,39)(27,36)(28,42)(38,47)(40,41)(43,48);(3,10)(6,13)(9,15)(12,16)(17,20)(18,26)(19,29)(21,35)(22,33)(23,34)(24,38)(25,39)(27,41)(28,43)(30,46)(31,44)(32,45)(36,40)(37,47)(42,48);(2,7)(6,13)(8,14)(12,16)(17,21,23,22)(18,27,25,36)(19,30,32,31)(20,33,34,35)(24,28,37,42)(26,40,39,41)(29,44,45,46)(38,48,47,43)Polycyclic presentation:
2^9|b^2=f,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=ei,d^c=di,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265173);