Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351720 |
| 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 | |
2685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,7)(3,11)(4,5)(9,10)(12,13)(14,15)(17,18)(19,29)(20,25)(21,28)(22,27)(23,26)(24,31)(30,44)(32,41)(33,42)(34,39)(35,38)(36,40)(37,46)(43,47)(45,48);(1,2)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,19,23,32)(18,24,26,38)(20,30,34,47)(21,31,22,35)(25,37,39,48)(27,29,28,41)(33,45,36,46)(40,43,42,44);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20)(18,25)(19,30)(21,33)(22,36)(23,34)(24,37)(26,39)(27,40)(28,42)(29,43)(31,45)(32,47)(35,46)(38,48)(41,44);(18,26)(24,38)(25,39)(27,28)(29,41)(37,48)(40,42)(43,44);(1,4,6,5)(2,9,10,7)(3,11,12,13)(8,15,16,14)(17,21,23,22)(18,27,26,28)(19,31,32,35)(20,33,34,36)(24,29,38,41)(25,40,39,42)(30,45,47,46)(37,43,48,44)Polycyclic presentation:
2^9|b^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258924);