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 | |
3964 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(33,34)(35,43)(36,40)(37,38)(39,42)(41,48)(44,45)(46,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(19,21)(24,26)(27,29)(31,32)(33,35,39,36)(34,40,42,43)(37,46,41,44)(38,45,48,47);(2,3)(7,9)(8,10)(14,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,34,39,42)(35,40,36,43)(37,38,41,48)(44,47,46,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36,39,35)(34,40,42,43)(37,46,41,44)(38,47,48,45);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,37)(34,38)(35,44)(36,46)(39,41)(40,47)(42,48)(43,45);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38)(34,41)(35,45)(36,47)(37,42)(39,48)(40,44)(43,46)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=h,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620658);