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 | |
3818 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)(18,22)(19,25)(24,29)(26,31)(33,34)(35,40)(36,42)(37,41)(38,43)(39,44)(45,46)(47,48);(2,7)(3,11)(9,13)(10,12)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,39,37)(34,40,44,41)(36,47,43,45)(38,46,42,48);(2,7)(3,11)(9,13)(10,12)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,39,37)(34,41,44,40)(36,45,43,47)(38,46,42,48);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,16)(10,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,43)(34,42,44,38)(35,45,37,47)(40,46,41,48);(1,4,6,15)(2,9,7,13)(3,12,11,10)(5,8,16,14)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37,39,35)(34,40,44,41)(36,47,43,45)(38,48,42,46);(1,5,6,16)(2,10,7,12)(3,9,11,13)(4,14,15,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,42)(34,43,44,36)(35,46,37,48)(40,45,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=h,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, 8646561);