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 | |
4207 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)(3,10,4,7)(5,8,14,13)(11,16,12,15)(17,18)(19,21)(20,29)(22,30)(23,25)(24,28)(26,27)(31,32)(33,34)(35,41)(36,40)(37,38)(39,42)(43,48)(44,47)(45,46);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,28)(18,24,25,21)(20,26,30,32)(22,27,29,31)(35,42)(39,41)(43,44)(47,48);(1,2,6,9)(3,7,4,10)(5,13,14,8)(11,16,12,15)(17,20,23,30)(18,22,25,29)(19,26,28,32)(21,31,24,27)(33,35,38,42)(34,39,37,41)(36,44,46,43)(40,48,45,47);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20)(18,22)(19,26)(21,31)(23,30)(24,27)(25,29)(28,32)(33,36,38,46)(34,40,37,45)(35,43,42,44)(39,47,41,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21,23,24)(18,19,25,28)(20,27,30,31)(22,32,29,26)(33,35)(34,41)(36,44)(37,39)(38,42)(40,47)(43,46)(45,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,22)(18,20)(19,27)(21,32)(23,29)(24,26)(25,30)(28,31)(33,37,38,34)(35,39,42,41)(36,45,46,40)(43,47,44,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gi,d^2=i,e^2=g,f^2=gh,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, 8900311);