Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351101 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2740 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)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,29)(20,26)(21,28)(22,25)(23,27)(24,31)(30,43)(32,39)(33,37)(34,42)(35,41)(36,40)(38,45)(44,47)(46,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,19,23,33)(18,24,27,39)(20,30,34,44)(21,32,22,31)(25,37,28,29)(26,38,42,48)(35,46,36,45)(40,47,41,43);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(18,25)(20,34)(21,22)(24,37)(26,41)(27,28)(29,39)(30,44)(31,32)(38,43)(40,42)(47,48);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(17,20,23,34)(18,26,27,42)(19,30,33,44)(21,35,22,36)(24,38,39,48)(25,40,28,41)(29,43,37,47)(31,45,32,46);(2,10)(7,16)(8,15)(9,14)(18,27)(20,34)(24,39)(25,28)(29,37)(30,44)(35,36)(45,46)Polycyclic presentation:
2^9|b^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258369);