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 | |
4177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(2,8)(4,9)(7,14)(10,15)(18,25)(19,29)(20,31)(27,32)(33,35,38,36)(34,40,42,39)(37,43,46,44)(41,48,45,47);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,36,38,35)(34,39,42,40)(37,44,46,43)(41,47,45,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,34)(35,40)(36,39)(37,45)(38,42)(41,46)(43,47)(44,48);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^9|b^2=ghi,c^2=gi,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9837448);