Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366710 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,8,13)(3,14,9,16)(4,10,5,15)(6,12,7,11)(18,24)(21,22)(25,38)(26,39)(27,28)(29,42)(31,36)(33,41)(34,37)(40,44)(43,48)(45,46);(1,3,8,9)(2,5,13,4)(6,10,7,15)(11,14,12,16)(17,18,23,28)(19,29,32,25)(20,33,35,39)(21,24,22,27)(26,37,41,34)(30,44,47,48)(31,38,36,42)(40,46,43,45);(1,4)(2,9)(3,13)(5,8)(6,14)(7,16)(10,11)(12,15)(17,19,23,32)(18,25,28,29)(20,30,35,47)(21,36,22,31)(24,38,27,42)(26,40,41,43)(33,48,39,44)(34,46,37,45);(1,5)(2,10)(3,12)(4,8)(6,14)(7,16)(9,11)(13,15)(17,20)(18,26)(19,30)(21,34)(22,37)(23,35)(24,39)(25,40)(27,33)(28,41)(29,43)(31,45)(32,47)(36,46)(38,44)(42,48);(1,6,8,7)(2,11,13,12)(3,10,9,15)(4,16,5,14)(17,21,23,22)(18,27,28,24)(19,31,32,36)(20,34,35,37)(25,38,29,42)(26,33,41,39)(30,45,47,46)(40,44,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273992);