Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354913 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,10)(3,5,9,6)(4,8,13,16)(11,12,15,14)(17,18,23,28)(19,29,33,39)(20,34,37,42)(21,26,22,27)(24,32,40,31)(25,35,41,36)(30,44,47,38)(43,45,48,46);(1,3,6,2,7,9,5,10)(4,11,14,8,13,15,12,16)(17,19,21,31,23,33,22,32)(18,24,27,29,28,40,26,39)(20,30,36,46,37,47,35,45)(25,38,34,43,41,44,42,48);(1,4)(2,8)(3,11)(5,12)(6,14)(7,13)(9,15)(10,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,42)(27,34)(28,41)(29,43)(31,46)(32,45)(33,47)(39,48)(40,44);(1,5,7,6)(2,9,10,3)(4,12,13,14)(8,15,16,11);(4,13)(8,16)(11,15)(12,14)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,39,40,29)(25,34,41,42)(30,45,47,46)(38,43,44,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^c=eh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262171);