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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,6,5,9)(4,10,15,8)(12,14,13,16)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(34,38)(35,43)(36,46)(42,48);(1,3,7,5)(2,6,11,9)(4,13,15,12)(8,14,10,16)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,34,39,38)(35,44,43,40)(36,41,46,47)(37,48,45,42);(1,4,7,15)(2,8,11,10)(3,12,5,13)(6,16,9,14)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,35,39,43)(34,40,38,44)(36,45,46,37)(41,48,47,42);(1,5,7,3)(2,9,11,6)(4,12,15,13)(8,16,10,14)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,4)(2,10)(3,12)(5,13)(6,14)(7,15)(8,11)(9,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,37,39,45)(34,42,38,48)(35,36,43,46)(40,41,44,47);(1,6,7,9)(2,3,11,5)(4,14,15,16)(8,13,10,12)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,38,39,34)(35,44,43,40)(36,47,46,41)(37,48,45,42)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900345);