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 | |
4293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,20)(12,17)(13,16)(14,18)(15,19)(21,22)(23,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,38)(36,39)(37,40);(1,2,5,4)(3,7,8,6)(9,11,15,13)(10,16,19,20)(12,21,18,23)(14,22,17,24)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,40,36,38);(1,3,5,8)(2,6,4,7)(26,28)(27,34)(33,36)(35,37);(1,4,5,2)(3,7,8,6)(9,12,15,18)(10,17,19,14)(11,21,13,23)(16,24,20,22)(25,28,31,26)(27,30,34,32)(29,36,39,33)(35,40,37,38);(9,13,15,11)(10,16,19,20)(12,23,18,21)(14,22,17,24)(25,29,31,39)(26,33,28,36)(27,35,34,37)(30,40,32,38);(9,14,15,17)(10,18,19,12)(11,22,13,24)(16,23,20,21)(25,30)(26,27)(28,34)(29,38)(31,32)(33,37)(35,36)(39,40)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=gh,e^2=gi,f^2=ghi,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, 9633715);