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 | |
4034 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)(3,11)(4,13)(5,12)(6,10)(7,8)(9,15)(14,16)(17,18,23,25)(19,24,29,32)(20,26,27,21)(22,28,31,30)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(1,3,7,12)(2,5,8,11)(4,14,15,6)(9,16,13,10)(17,19,23,29)(18,24,25,32)(20,28,27,30)(21,22,26,31)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,8)(5,11)(9,13)(10,16)(18,25)(19,29)(21,26)(28,30)(35,40)(37,45)(43,46)(44,48);(1,4,7,15)(2,9,8,13)(3,6,12,14)(5,10,11,16)(17,20,23,27)(18,26,25,21)(19,30,29,28)(22,32,31,24)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,5)(2,3)(4,10)(6,9)(7,11)(8,12)(13,14)(15,16)(17,21,23,26)(18,27,25,20)(19,31,29,22)(24,28,32,30)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,6)(2,10)(3,4)(5,9)(7,14)(8,16)(11,13)(12,15)(17,22,23,31)(18,28,25,30)(19,26,29,21)(20,24,27,32)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=h,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529451);