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 | |
4101 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)(3,4,9,8)(5,10,15,16)(11,12,13,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(2,7)(3,9)(10,16)(11,13)(18,19)(23,25)(24,26)(30,31)(35,44)(36,45)(40,46)(41,48);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35)(34,40)(36,37)(38,41)(39,44)(42,45)(43,46)(47,48);(1,3,6,9)(2,8,7,4)(5,13,15,11)(10,12,16,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,41)(35,37)(38,40)(39,45)(42,44)(43,48)(46,47);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,16,13)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37)(34,38)(35,36)(39,42)(40,41)(43,47)(44,45)(46,48);(1,5,6,15)(2,10,7,16)(3,11,9,13)(4,12,8,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38)(34,42)(35,41)(36,46)(37,43)(39,47)(40,45)(44,48)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8598227);