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 | |
4170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,12,13,9)(11,16,14,15)(20,27)(24,31)(26,28)(29,32)(33,34,38,39)(35,40,46,41)(36,37,43,42)(44,45,47,48);(2,7)(4,13)(5,8)(11,14)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,31)(28,32)(34,39)(37,42)(40,41)(45,48);(2,7)(3,10)(9,12)(11,14)(18,23)(19,25)(24,31)(26,28)(33,35,38,46)(34,40,39,41)(36,44,43,47)(37,45,42,48);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,19,22,25)(18,21,23,30)(20,28,27,26)(24,32,31,29)(33,35)(34,41)(36,47)(37,45)(38,46)(39,40)(42,48)(43,44);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,20,22,27)(18,24,23,31)(19,26,25,28)(21,29,30,32)(33,36,38,43)(34,42,39,37)(35,44,46,47)(40,48,41,45);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,37,38,42)(34,43,39,36)(35,45,46,48)(40,47,41,44)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8769666);