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 | |
4394 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)(3,4)(5,7)(6,8)(9,10,15,17)(11,21,22,16)(12,13,19,18)(14,24,23,20)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(1,3,6,7)(2,5,8,4)(9,11)(10,16)(12,23)(13,24)(14,19)(15,22)(17,21)(18,20)(27,33)(31,34)(35,37)(39,40);(10,17)(11,22)(12,19)(20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(1,4,6,5)(2,7,8,3)(9,12,15,19)(10,18,17,13)(11,23,22,14)(16,20,21,24)(25,28,30,26)(27,34,33,31)(29,36,38,32)(35,40,37,39);(1,5,6,4)(2,7,8,3)(9,13,15,18)(10,19,17,12)(11,20,22,24)(14,16,23,21)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(1,2)(3,5)(4,7)(6,8)(9,14,15,23)(10,20,17,24)(11,19,22,12)(13,16,18,21)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=ghi,e^2=g,f^2=h,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, 9722297);