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 | |
5011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(1,3,5,8)(2,6,7,4)(10,14)(15,17)(16,21)(22,23)(28,35)(32,38)(34,39)(36,40);(1,3)(2,4)(5,8)(6,7)(12,20)(16,21)(18,24)(22,23)(26,31)(27,33)(32,38)(34,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,29)(28,34)(30,33)(31,37)(32,36)(35,39)(38,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29)(26,27)(28,36)(30,37)(31,33)(32,34)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=hi,b^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150330);