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 | |
4176 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)(3,8,11,14)(4,13,15,9)(5,10,16,12)(17,18)(19,22)(20,25)(21,24)(23,28)(26,30)(27,32)(29,31)(33,34)(35,38)(36,40)(37,41)(39,42)(43,46)(44,45)(47,48);(2,7)(3,11)(9,13)(10,12)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)(33,35)(34,38)(36,43)(37,42)(39,41)(40,46)(44,47)(45,48);(2,7)(3,11)(9,13)(10,12)(18,19)(23,25)(24,26)(31,32)(35,42)(38,39)(43,48)(46,47);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,16)(10,15)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,46)(45,48);(1,4,6,15)(2,9,7,13)(3,12,11,10)(5,8,16,14)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,5,6,16)(2,10,7,12)(3,9,11,13)(4,14,15,8)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,34)(35,38)(36,44)(37,41)(39,42)(40,45)(43,47)(46,48)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635419);