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 | |
3765 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)(3,4,9,12)(5,8,15,16)(10,11,13,14)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32)(33,34,37,40)(35,38,42,46)(36,43,44,39)(41,47,48,45);(2,7)(3,9)(8,16)(10,13)(18,23)(19,26)(25,28)(27,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,7)(3,9)(8,16)(10,13)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,4)(5,13)(6,9)(7,12)(8,14)(10,15)(11,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,4)(2,3)(5,14)(6,12)(7,9)(8,13)(10,16)(11,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47);(1,5,6,15)(2,8,7,16)(3,10,9,13)(4,11,12,14)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,d^2=i,e^2=gh,f^2=gi,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, 8658645);