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 | |
4437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,39)(38,40);(1,3)(2,4)(5,6)(7,8)(12,19)(16,22)(18,23)(20,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,7,6)(3,5,8,4)(10,15)(11,17)(16,22)(18,23)(26,27)(31,33)(32,34)(38,39);(1,4,7,5)(2,8,6,3)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,5,7,4)(2,8,6,3)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,6,7,2)(3,5,8,4)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279, #72219281 |
| maximal quotient of | #72219276 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620651);