Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351885 |
| 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 | |
2639 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)(18,24)(19,28)(21,22)(26,37)(27,34)(29,42)(30,35)(32,33)(36,39)(41,44)(43,45)(46,48);(1,2,6,9)(3,7,11,16)(4,8,5,12)(10,15,13,14)(17,18)(19,25)(20,26)(21,34)(22,24)(23,27)(28,39)(29,38)(30,40)(31,41)(32,44)(33,37)(35,36)(42,48)(43,47)(45,46);(1,3)(2,7)(4,10)(5,13)(6,11)(8,15)(9,16)(12,14)(17,19,23,30)(18,25,27,40)(20,29,31,43)(21,28,22,35)(24,36,34,39)(26,38,41,47)(32,42,33,45)(37,46,44,48);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,15,16,14)(17,20)(18,26)(19,29)(21,32)(22,33)(23,31)(24,37)(25,38)(27,41)(28,42)(30,43)(34,44)(35,45)(36,46)(39,48)(40,47);(20,31)(26,41)(29,43)(32,33)(37,44)(38,47)(42,45)(46,48)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=i,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259171);