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) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,6)(3,5,8,4)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,31,28)(27,29,32,34)(30,36,39,33)(35,38,40,37);(1,3)(2,4)(5,6)(7,8)(12,19)(16,22)(18,23)(20,24)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,38)(39,40);(10,15)(11,17)(16,22)(18,23)(26,28)(27,32)(33,36)(35,40);(1,4,7,5)(2,8,6,3)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28,31,26)(27,34,32,29)(30,36,39,33)(35,38,40,37);(1,5,7,4)(2,8,6,3)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,34)(26,32,28,27)(30,38,39,37)(33,35,36,40);(1,6,7,2)(3,5,8,4)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^9|a^2=gh,d^2=gi,e^2=gi,f^2=g,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, 9765720);