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 | |
4459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(25,26,30,32)(27,31,33,28)(29,34,38,39)(35,37,40,36);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,37,39,36);(10,17)(12,22)(16,19)(20,23)(26,32)(27,33)(34,39)(35,40);(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,40)(38,39);(1,3,4,7)(2,5,6,8)(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,28,30,31)(26,33,32,27)(29,37,38,36)(34,35,39,40);(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=gi,b^2=h,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7829650);