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 | |
4505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,16,21,23)(12,13,18,17)(19,24,22,20)(25,26,30,34)(27,28,32,31)(29,33,37,40)(35,36,39,38);(1,3,5,8)(2,6,7,4)(10,15)(12,18)(16,23)(19,22)(25,27,30,32)(26,31,34,28)(29,35,37,39)(33,38,40,36);(1,3)(2,4)(5,8)(6,7)(10,15)(13,17)(16,23)(20,24)(29,37)(33,40)(35,39)(36,38);(1,4)(2,3)(5,6)(7,8)(9,11,14,21)(10,16,15,23)(12,22,18,19)(13,20,17,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(9,12,14,18)(10,17,15,13)(11,19,21,22)(16,20,23,24)(25,28)(26,32)(27,34)(29,36)(30,31)(33,39)(35,40)(37,38);(9,13,14,17)(10,18,15,12)(11,20,21,24)(16,19,23,22)(25,29,30,37)(26,33,34,40)(27,35,32,39)(28,36,31,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325335);