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 | |
4198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,8,10,7)(4,13,14,5)(11,12,16,15)(18,21)(19,20)(26,29)(27,28)(33,34,40,39)(35,44,41,36)(37,46,48,42)(38,45,47,43);(3,10)(7,8)(11,16)(12,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,35,40,41)(34,36,39,44)(37,43,48,45)(38,46,47,42);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,15)(14,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,36)(34,41)(35,39)(37,47)(38,48)(40,44)(42,43)(45,46);(1,3)(2,8)(4,11)(5,15)(6,10)(7,9)(12,13)(14,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,37,40,48)(34,42,39,46)(35,45,41,43)(36,47,44,38);(1,4)(2,5)(3,11)(6,14)(7,12)(8,15)(9,13)(10,16)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,38,40,47)(34,43,39,45)(35,46,41,42)(36,48,44,37);(1,5)(2,4)(3,12)(6,13)(7,11)(8,16)(9,14)(10,15)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,39,40,34)(35,44,41,36)(37,42,48,46)(38,45,47,43)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332564);