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 | |
4028 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)(3,7,12,5)(4,8,14,16)(10,15,11,13)(17,18,23,20)(19,25,21,24)(22,28,30,26)(27,31,29,32)(34,39)(36,43)(41,45)(44,48);(1,3,6,12)(2,7,9,5)(4,11,14,10)(8,13,16,15)(17,18,23,20)(19,25,21,24)(22,26,30,28)(27,32,29,31)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,46)(47,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,38,40)(34,36,39,43)(37,42,46,47)(41,44,45,48);(1,4)(2,8)(3,11)(5,15)(6,14)(7,13)(9,16)(10,12)(17,20,23,18)(19,25,21,24)(22,26,30,28)(27,31,29,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,48)(44,47);(1,3)(2,5)(4,10)(6,12)(7,9)(8,13)(11,14)(15,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,36)(34,40)(35,39)(37,44)(38,43)(41,47)(42,45)(46,48);(1,5)(2,3)(4,13)(6,7)(8,10)(9,12)(11,16)(14,15)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,37,38,46)(34,41,39,45)(35,42,40,47)(36,44,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329897);