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 | |
4196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,9)(6,10,14,15)(11,13,12,16)(20,28)(24,31)(27,32)(29,30)(33,34)(35,44)(36,41)(37,45)(38,46)(39,42)(40,43)(47,48);(2,8)(3,4)(10,15)(11,12)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,40,42)(34,39,43,44)(36,45,47,38)(37,41,46,48);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36,40,47)(34,41,43,48)(35,45,42,38)(37,39,46,44);(1,4,7,3)(2,5,8,9)(6,12,14,11)(10,13,15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,37)(34,38)(35,41)(36,39)(40,46)(42,48)(43,45)(44,47);(1,5)(2,3)(4,8)(6,13)(7,9)(10,11)(12,15)(14,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,38)(34,37)(35,36)(39,41)(40,45)(42,47)(43,46)(44,48);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,39)(34,42)(35,43)(36,46)(37,47)(38,41)(40,44)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8366292);