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 |
| prime radical | #35772 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,19)(11,18,21,13)(12,22,23,16)(14,17,24,20)(25,26,30,34)(27,35,36,31)(28,32,29,33)(37,40,38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,20)(14,16)(15,21)(17,23)(18,19)(22,24)(27,36)(31,35)(37,38)(39,40);(9,12,15,23)(10,16,19,22)(11,20,21,17)(13,14,18,24)(25,27,30,36)(26,31,34,35)(28,37,29,38)(32,39,33,40);(1,3,4,7)(2,6,5,8)(9,13,15,18)(10,11,19,21)(12,14,23,24)(16,20,22,17)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,39)(38,40);(9,14)(10,17)(11,16)(12,18)(13,23)(15,24)(19,20)(21,22)(25,28,30,29)(26,32,34,33)(27,37,36,38)(31,39,35,40);(9,11)(10,18)(12,20)(13,19)(14,22)(15,21)(16,24)(17,23)(25,29,30,28)(26,33,34,32)(27,38,36,37)(31,40,35,39)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565755);