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 | |
4527 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)(25,26)(27,28)(29,33)(30,32)(31,34)(35,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27)(26,31)(28,32)(29,37)(30,34)(33,36)(35,38)(39,40);(1,3,4,7)(2,6,5,8)(10,14)(11,17)(16,19)(18,23)(26,32)(27,34)(33,40)(35,37);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,30,31)(26,27,32,34)(29,36,38,39)(33,35,40,37);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,27)(26,31)(28,32)(29,35)(30,34)(33,39)(36,40)(37,38);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,30,38)(26,33,32,40)(27,35,34,37)(28,36,31,39)Polycyclic presentation:
2^9|c^2=gh,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644604);