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 | |
4373 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(25,26)(27,36)(28,38)(29,33)(30,37)(31,32)(34,39)(35,40);(2,5)(6,7)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(26,32)(27,29)(34,38)(35,37);(10,16)(12,17)(15,22)(18,23)(25,27,31,29)(26,33,32,36)(28,35,39,37)(30,34,40,38);(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,28,31,39)(26,34,32,38)(27,37,29,35)(30,36,40,33);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,29,31,27)(26,33,32,36)(28,35,39,37)(30,38,40,34);(1,3,4,8)(2,6,5,7)(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,30)(26,35)(27,34)(28,36)(29,38)(31,40)(32,37)(33,39)Polycyclic presentation:
2^9|b^2=hi,c^2=i,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=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219281, #72219283 |
| maximal quotient of | #72219276 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9837450);