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 | |
4317 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)(3,9)(4,5)(6,14)(7,8)(10,15)(11,12)(13,16)(21,29)(24,30)(26,31)(27,32)(33,34)(35,41)(36,45)(37,40)(38,47)(39,42)(43,44)(46,48);(1,3,7,5)(2,4,8,9)(6,11,15,13)(10,12,14,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(33,35,39,37)(34,40,42,41)(36,43,47,46)(38,44,45,48);(2,8)(4,9)(10,14)(12,16)(18,23)(19,25)(24,30)(26,31)(33,35,39,37)(34,41,42,40)(36,46,47,43)(38,44,45,48);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,14)(15,16)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,36)(34,38)(35,43)(37,46)(39,47)(40,44)(41,48)(42,45);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,37,39,35)(34,40,42,41)(36,43,47,46)(38,48,45,44);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38)(34,36)(35,44)(37,48)(39,45)(40,43)(41,46)(42,47)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066599);