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 | |
4138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(9,14)(11,16)(17,18)(19,27)(20,31)(21,26)(22,23)(24,30)(25,29)(28,32)(33,34,39,36)(35,40,38,43)(37,41,45,44)(42,47,46,48);(1,2,6,7)(3,4,10,8)(5,9,13,14)(11,12,16,15)(17,19,23,30)(18,24,22,27)(20,21,29,28)(25,26,31,32)(33,35,39,38)(34,40,36,43)(37,42,45,46)(41,47,44,48);(3,10)(4,8)(5,13)(9,14)(18,22)(20,29)(21,28)(24,27)(34,36)(35,38)(37,45)(47,48);(1,3,6,10)(2,8,7,4)(5,11,13,16)(9,15,14,12)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,32)(26,27)(33,36,39,34)(35,40,38,43)(37,41,45,44)(42,48,46,47);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,31,27,25)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,32,28,26)(33,38,39,35)(34,40,36,43)(37,46,45,42)(41,47,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969326);