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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18,24,21)(19,27,23,25)(20,29,22,26)(28,31,30,32)(34,39)(35,42)(37,46)(44,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,24,23)(18,25,21,27)(20,28,22,30)(26,31,29,32)(34,39)(36,41)(40,43)(44,47);(2,9)(3,4)(10,16)(11,12)(17,20,24,22)(18,26,21,29)(19,28,23,30)(25,31,27,32)(33,34,38,39)(35,43,42,40)(36,37,41,46)(44,48,47,45);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,21,24,18)(19,27,23,25)(20,26,22,29)(28,31,30,32)(33,35,38,42)(34,40,39,43)(36,44,41,47)(37,45,46,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,22,24,20)(18,26,21,29)(19,28,23,30)(25,32,27,31)(33,36)(34,37)(35,44)(38,41)(39,46)(40,45)(42,47)(43,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,23,24,19)(18,27,21,25)(20,30,22,28)(26,32,29,31)(33,37)(34,41)(35,45)(36,39)(38,46)(40,47)(42,48)(43,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620683);