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 | #2388 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,11,5)(4,12,14,7)(10,13,16,15)(18,23)(19,26)(20,28)(29,31)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(1,3,6,11)(2,5,8,9)(4,10,14,16)(7,13,12,15)(17,18,22,23)(19,21,26,24)(20,25,28,30)(27,29,32,31)(34,39)(35,41)(40,46)(42,47);(1,2)(3,5)(4,7)(6,8)(9,11)(10,13)(12,14)(15,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,2)(3,9)(4,7)(5,11)(6,8)(10,15)(12,14)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,4,6,14)(2,7,8,12)(3,10,11,16)(5,13,9,15)(17,20,22,28)(18,25,23,30)(19,27,26,32)(21,29,24,31);(1,5,6,9)(2,3,8,11)(4,13,14,15)(7,10,12,16)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332817);