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 | |
4744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,21)(14,15)(16,23)(17,20)(18,22)(19,24)(25,26)(27,36)(28,37)(29,35)(30,32)(31,33)(34,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,17)(10,12,15,20)(13,18,23,24)(16,19,21,22)(25,27,31,30)(26,32,33,36)(28,35,38,39)(29,37,40,34);(1,3,5,6)(2,4,7,8)(10,15)(11,17)(16,21)(18,24)(26,33)(28,38)(32,36)(35,39);(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,28)(26,34)(27,35)(29,32)(30,39)(31,38)(33,37)(36,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,22)(14,20)(15,17)(16,24)(18,21)(19,23)(25,29)(26,35)(27,34)(28,32)(30,37)(31,40)(33,39)(36,38);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,18)(12,19)(14,23)(15,21)(17,24)(20,22)(25,30,31,27)(26,36,33,32)(28,39,38,35)(29,34,40,37)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9588852);