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 | |
4130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(17,18)(19,24)(20,21)(22,30)(23,25)(26,32)(27,29)(28,31)(35,41)(38,44)(42,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,23,28)(18,24,25,31)(20,22,26,29)(21,30,32,27)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(3,9)(7,12)(10,15)(14,16)(18,25)(19,28)(20,26)(27,30)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,23,26)(18,21,25,32)(19,29,28,22)(24,27,31,30)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21)(18,26)(19,30)(20,25)(22,31)(23,32)(24,29)(27,28)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,22,23,29)(18,27,25,30)(19,26,28,20)(21,31,32,24)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48)Polycyclic presentation:
2^9|b^2=h,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014520);