Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349830 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,10)(4,5)(7,14)(8,12)(11,13)(15,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,42)(30,38)(32,41)(33,39)(34,40)(35,46)(37,45)(43,47)(44,48);(1,2)(3,7)(4,8)(5,12)(6,9)(10,15)(11,16)(13,14)(17,19,23,32)(18,24,27,38)(20,29,34,44)(21,31,22,30)(25,37,40,47)(26,41,36,28)(33,45,35,43)(39,48,46,42);(1,3)(2,7)(4,10)(5,13)(6,11)(8,15)(9,16)(12,14)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,39)(27,40)(28,42)(30,43)(31,45)(32,44)(36,46)(38,47)(41,48);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,15,16,14)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,41)(25,39,40,46)(29,43,44,45)(37,42,47,48);(20,34)(25,40)(29,44)(33,35)(37,47)(39,46)(42,48)(43,45)Polycyclic presentation:
2^9|b^2=h,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,d^b=dh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257051);