Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,13,16)(4,14,15,10)(5,11,12,6)(17,18,23,21)(19,26,22,24)(20,25,30,29)(27,32,28,31)(33,34,39,43)(35,41,45,37)(36,40,46,48)(38,42,47,44);(1,3)(2,8)(4,6)(5,14)(7,13)(9,16)(10,12)(11,15)(17,19,23,22)(18,24,21,26)(20,28,30,27)(25,32,29,31)(33,35,39,45)(34,37,43,41)(36,44,46,42)(38,40,47,48);(2,9)(3,13)(4,15)(5,12)(17,19,23,22)(18,24,21,26)(20,28,30,27)(25,32,29,31)(33,36,39,46)(34,40,43,48)(35,44,45,42)(37,47,41,38);(1,4)(2,10)(3,11)(5,8)(6,13)(7,15)(9,14)(12,16)(17,20,23,30)(18,25,21,29)(19,27,22,28)(24,31,26,32)(33,35)(34,41)(36,44)(37,43)(38,40)(39,45)(42,46)(47,48);(1,5)(2,11)(3,10)(4,8)(6,9)(7,12)(13,14)(15,16)(17,21,23,18)(19,24,22,26)(20,29,30,25)(27,31,28,32)(33,37,39,41)(34,35,43,45)(36,38,46,47)(40,42,48,44);(1,6,7,11)(2,12,9,5)(3,4,13,15)(8,10,16,14)(17,22,23,19)(18,26,21,24)(20,28,30,27)(25,32,29,31)(33,38)(34,42)(35,40)(36,37)(39,47)(41,46)(43,44)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947785);