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 | |
4406 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)(11,18)(12,14)(13,16)(15,17)(19,24)(20,23)(21,22)(25,26)(27,31)(28,36)(29,34)(30,32)(33,38)(35,40)(37,39);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,20)(13,15)(14,21)(17,18)(19,22)(23,24)(25,27,30,29)(26,31,32,34)(28,35,38,37)(33,39,36,40);(10,17)(11,13)(12,19)(21,23)(26,32)(31,34)(33,36)(39,40);(1,3,4,7)(2,6,5,8)(9,12,15,19)(10,14,17,24)(11,20,13,22)(16,21,18,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,38)(37,40);(9,13,15,11)(10,18,17,16)(12,22,19,20)(14,23,24,21)(25,28,30,38)(26,33,32,36)(27,35,29,37)(31,39,34,40);(9,14)(10,19)(11,21)(12,17)(13,23)(15,24)(16,22)(18,20)(25,29,30,27)(26,34,32,31)(28,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|b^2=gh,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7627430);