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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(25,26)(27,31)(28,37)(29,39)(30,32)(33,38)(34,36)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,27,30,35)(26,31,32,40)(28,36,38,29)(33,39,37,34);(10,17)(12,20)(16,19)(21,23)(26,32)(27,35)(28,38)(34,39);(1,3,4,7)(2,6,5,8)(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,39)(37,38);(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,28)(26,33)(27,36)(29,35)(30,38)(31,39)(32,37)(34,40);(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,29,30,36)(26,34,32,39)(27,28,35,38)(31,33,40,37)Polycyclic presentation:
2^9|b^2=h,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618296);