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 | #46248 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3839 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)(3,11,12,8)(4,14,15,9)(5,13,16,10)(17,18,23,24)(19,27,28,25)(20,31,32,21)(22,29,30,26)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,7)(5,16)(8,11)(9,14)(18,24)(19,28)(21,31)(26,29)(36,44)(39,46)(42,48)(45,47);(2,7)(3,12)(9,14)(10,13)(17,19,23,28)(18,25,24,27)(20,29,32,26)(21,30,31,22)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,6,12)(2,8,7,11)(4,13,15,10)(5,9,16,14)(17,20)(18,21)(19,29)(22,27)(23,32)(24,31)(25,30)(26,28);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21)(18,20)(19,30)(22,28)(23,31)(24,32)(25,29)(26,27)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,22,23,30)(18,26,24,29)(19,21,28,31)(20,27,32,25)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662285);