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 | |
3918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,8)(5,10,15,16)(11,12,13,14)(33,34,40,39)(35,42,36,41)(37,44,38,43)(45,47,46,48);(2,7)(3,9)(10,16)(11,13)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,35,40,36)(34,41,39,42)(37,45,38,46)(43,47,44,48);(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36,40,35)(34,42,39,41)(37,46,38,45)(43,48,44,47);(1,3,6,9)(2,8,7,4)(5,13,15,11)(10,12,16,14)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,37,40,38)(34,43,39,44)(35,45,36,46)(41,47,42,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,16,13)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,38,40,37)(34,44,39,43)(35,46,36,45)(41,48,42,47);(1,5,6,15)(2,10,7,16)(3,11,9,13)(4,12,8,14)(17,21,22,18)(19,25,27,23)(20,24,29,28)(26,30,32,31)(33,39,40,34)(35,42,36,41)(37,43,38,44)(45,47,46,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8608557);