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 | |
4213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,11,5)(4,8,14,16)(10,15,12,13)(17,18)(19,21)(20,29)(22,30)(23,25)(24,28)(26,27)(31,32)(34,36)(39,43)(40,44)(47,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,14)(13,16)(17,19,23,28)(18,24,25,21)(20,26,30,32)(22,27,29,31)(33,34,38,36)(35,39,42,43)(37,40,46,44)(41,47,45,48);(1,3,6,11)(2,7,9,5)(4,12,14,10)(8,13,16,15)(17,20,23,30)(18,22,25,29)(19,26,28,32)(21,31,24,27)(33,35,38,42)(34,39,36,43)(37,45,46,41)(40,48,44,47);(1,3)(2,7)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,20)(18,22)(19,26)(21,31)(23,30)(24,27)(25,29)(28,32)(33,36,38,34)(35,39,42,43)(37,44,46,40)(41,47,45,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,21,23,24)(18,19,25,28)(20,27,30,31)(22,32,29,26)(33,35)(34,39)(36,43)(37,41)(38,42)(40,47)(44,48)(45,46);(1,5,6,7)(2,3,9,11)(4,13,14,15)(8,10,16,12)(17,22)(18,20)(19,27)(21,32)(23,29)(24,26)(25,30)(28,31)(33,37)(34,40)(35,41)(36,44)(38,46)(39,47)(42,45)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8934530);