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 | |
4424 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)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26)(27,30)(28,29)(31,32)(33,37)(34,40)(35,38)(36,39);(1,2,5,4)(3,7,8,6)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27,31,34)(26,30,32,40)(28,35,37,39)(29,38,33,36);(1,3,5,8)(2,6,4,7)(10,15)(14,21)(16,19)(22,23)(26,32)(27,34)(28,37)(36,38);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,28)(26,33)(27,35)(29,32)(30,36)(31,37)(34,39)(38,40);(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,29,31,33)(26,28,32,37)(27,36,34,38)(30,39,40,35);(9,12)(10,16)(11,17)(13,20)(14,22)(15,19)(18,24)(21,23)(25,30)(26,34)(27,32)(28,36)(29,35)(31,40)(33,39)(37,38)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9624040);