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 | |
4162 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18)(19,27)(20,29)(21,24)(22,28)(23,25)(26,31)(30,32)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,26,32,29)(34,40)(35,36)(42,45)(43,44);(4,13)(8,12)(11,14)(15,16)(18,25)(20,31)(24,27)(28,30)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,4,6,13)(2,8,9,12)(3,11,5,14)(7,15,10,16)(17,20)(18,26)(19,28)(21,30)(22,24)(23,31)(25,29)(27,32)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,26,32,29)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,22,23,32)(18,28,25,30)(19,26,21,29)(20,27,31,24)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8628040);