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 | |
4292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,5,4)(6,10,15,16)(11,14,13,12)(17,18,22,25)(19,26,21,23)(20,24,28,31)(27,32,29,30)(34,40)(35,36)(37,45)(46,48);(1,3,7,5)(2,8,9,4)(6,11,15,13)(10,14,16,12)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,34)(35,41)(36,38)(37,42)(39,40)(43,48)(44,46)(45,47);(2,9)(3,5)(10,16)(11,13)(20,28)(24,31)(27,29)(30,32)(33,35,39,36)(34,41,40,38)(37,44,45,43)(42,46,47,48);(1,4,7,8)(2,3,9,5)(6,14,15,12)(10,13,16,11)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,36,39,35)(34,41,40,38)(37,44,45,43)(42,48,47,46);(1,5,7,3)(2,8,9,4)(6,13,15,11)(10,14,16,12)(17,20,22,28)(18,24,25,31)(19,27,21,29)(23,30,26,32)(33,37,39,45)(34,42,40,47)(35,43,36,44)(38,46,41,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)(33,38,39,41)(34,36,40,35)(37,46,45,48)(42,44,47,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441864);