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 | |
3836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(34,39)(35,40)(41,46)(42,47);(2,7)(5,14)(8,11)(12,16)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8134575);