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 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,9,4,8)(5,13,14,10)(11,15,12,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34)(35,36)(37,40)(38,45)(39,41)(42,47)(43,48)(44,46);(1,3,7,4)(2,8,6,9)(5,11,14,12)(10,15,13,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,40,39)(34,41,37,36)(38,43,47,46)(42,48,45,44);(2,6)(8,9)(10,13)(15,16)(22,30)(25,28)(27,29)(31,32)(33,36)(34,39)(35,37)(38,44)(40,41)(42,46)(43,45)(47,48);(1,4,7,3)(2,9,6,8)(5,12,14,11)(10,16,13,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,37,40,34)(35,41,39,36)(38,45,47,42)(43,48,46,44);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,39,40,35)(34,41,37,36)(38,46,47,43)(42,48,45,44)Polycyclic presentation:
2^9|a^2=hi,b^2=g,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087580);