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 | |
4309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,16)(11,15,12,17)(13,21,22,18)(19,24,20,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,36)(39,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,14)(13,19)(16,17)(18,23)(20,22)(21,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(10,16)(11,12)(18,21)(19,20)(27,31)(28,34)(35,39)(36,40);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,28,30,34)(26,31,33,27)(29,36,38,40)(32,39,37,35);(9,13,14,22)(10,18,16,21)(11,19,12,20)(15,23,17,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621668);