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 | |
3879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,9,4,5)(6,10,15,16)(11,14,12,13)(18,24)(20,25)(26,32)(27,30);(2,8)(3,4)(10,16)(11,12)(17,18,23,24)(19,25,29,20)(21,26,28,32)(22,27,31,30)(34,38)(39,45)(40,46)(47,48);(1,3,7,4)(2,5,8,9)(6,11,15,12)(10,13,16,14)(17,19,23,29)(18,25,24,20)(21,22,28,31)(26,27,32,30)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,4,7,3)(2,9,8,5)(6,12,15,11)(10,14,16,13)(17,20)(18,19)(21,30)(22,26)(23,25)(24,29)(27,28)(31,32);(1,5,7,9)(2,3,8,4)(6,14,15,13)(10,12,16,11)(17,21)(18,26)(19,22)(20,30)(23,28)(24,32)(25,27)(29,31)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,6,7,15)(2,10,8,16)(3,11,4,12)(5,13,9,14)(17,22)(18,27)(19,28)(20,26)(21,29)(23,31)(24,30)(25,32)(33,36,37,44)(34,40,38,46)(35,41,42,43)(39,47,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003870);