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 | |
4550 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,21)(14,15)(16,18)(17,22)(19,23)(20,24)(25,26,31,28)(27,29,32,35)(30,33,39,36)(34,37,38,40);(2,6)(3,7)(10,15)(11,18)(13,22)(20,23)(25,27,31,32)(26,29,28,35)(30,38,39,34)(33,40,36,37);(1,3,5,7)(2,4,6,8)(10,15)(12,16)(17,21)(20,23)(27,32)(29,35)(34,38)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,21)(22,24)(25,28,31,26)(27,29,32,35)(30,36,39,33)(34,37,38,40);(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29)(26,32)(27,28)(30,37)(31,35)(33,38)(34,36)(39,40);(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,38)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10268120);