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 | |
3875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,26)(20,25)(21,22)(23,27)(24,31)(28,30)(29,32)(33,34,39,36)(35,40,45,43)(37,41,44,46)(38,42,48,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,35,39,45)(34,40,36,43)(37,38,44,48)(41,42,46,47);(2,9)(3,4)(10,16)(11,12)(17,20,23,21)(18,25,27,22)(19,28,31,29)(24,32,26,30)(33,34,39,36)(35,43,45,40)(37,46,44,41)(38,42,48,47);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,32,26,30)(33,36,39,34)(35,43,45,40)(37,41,44,46)(38,42,48,47);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,19)(18,26)(20,29)(21,28)(22,30)(23,31)(24,27)(25,32)(33,37,39,44)(34,41,36,46)(35,38,45,48)(40,42,43,47);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22,23,25)(18,21,27,20)(19,30,31,32)(24,29,26,28)(33,38)(34,42)(35,44)(36,47)(37,45)(39,48)(40,46)(41,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575485);