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 | |
4462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,21)(14,15)(16,22)(17,19)(18,24)(20,23)(25,26)(27,34)(28,33)(29,31)(30,32)(35,40)(36,39)(37,38);(1,3,5,8)(2,6,4,7)(10,15)(12,19)(16,21)(20,24)(25,27,30,29)(26,31,32,34)(28,36,38,35)(33,40,37,39);(1,2,5,4)(3,6,8,7)(10,15)(11,17)(16,21)(18,23)(26,32)(27,29)(33,37)(35,36);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,24)(22,23)(25,26)(27,31)(28,37)(29,34)(30,32)(33,38)(35,40)(36,39);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,23)(22,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,39)(32,37)(34,40);(9,13,14,22)(10,16,15,21)(11,18,17,23)(12,20,19,24)(25,29,30,27)(26,34,32,31)(28,36,38,35)(33,40,37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7908192);