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 | |
3795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,9,12,14)(11,16,13,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,7,5)(2,8,6,10)(4,11,12,13)(9,15,14,16)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(4,12)(9,14)(11,13)(15,16)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,7,3)(2,8,6,10)(4,13,12,11)(9,15,14,16)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,6,7,2)(3,10,5,8)(4,9,12,14)(11,15,13,16)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,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, 8658009);