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 | |
4391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26)(27,32)(28,29)(30,39)(31,33)(34,36)(35,40)(37,38);(1,2,5,4)(3,7,8,6)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,31,35)(26,32,33,40)(28,36,37,30)(29,34,38,39);(1,3,5,8)(2,6,4,7)(10,12)(15,17)(16,20)(22,23)(26,33)(27,35)(28,37)(34,39);(1,4,5,2)(3,7,8,6)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28)(26,29)(27,36)(30,35)(31,37)(32,34)(33,38)(39,40);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,29)(26,28)(27,34)(30,40)(31,38)(32,36)(33,37)(35,39);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30,31,36)(26,34,33,39)(27,28,35,37)(29,32,38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9722336);