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 | |
4135 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)(3,8)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(17,18,23,24)(19,25,20,21)(22,30,31,26)(27,29,28,32)(33,34,38,39)(35,40,43,36)(37,45,46,41)(42,44,48,47);(3,11)(4,13)(8,14)(9,16)(17,19)(18,21)(20,23)(22,27)(24,25)(26,29)(28,31)(30,32)(34,39)(36,40)(41,45)(44,47);(2,7)(3,11)(9,16)(10,12)(18,24)(19,20)(26,30)(27,28)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,47,45,44);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,32,30,29)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,47)(46,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(17,21,23,25)(18,19,24,20)(22,29,31,32)(26,27,30,28)(33,36)(34,35)(37,44)(38,40)(39,43)(41,42)(45,48)(46,47);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,22,23,31)(18,26,24,30)(19,27,20,28)(21,29,25,32)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000685);