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 | #58197 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,22)(12,17)(13,23)(14,19)(15,20)(16,21)(18,24)(27,32)(29,34)(35,36)(37,39);(1,2,4,6)(3,5,8,7)(9,11)(10,12)(13,19)(14,18)(15,24)(16,17)(20,23)(21,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,12,16,22)(10,17,21,11)(13,14,24,20)(15,23,19,18)(25,27,30,32)(26,29,33,34)(28,36,38,35)(31,39,40,37);(9,13,16,24)(10,18,21,23)(11,19,17,15)(12,14,22,20)(25,28,30,38)(26,31,33,40)(27,35,32,36)(29,37,34,39);(1,2)(3,7)(4,6)(5,8)(9,14)(10,19)(11,23)(12,24)(13,22)(15,21)(16,20)(17,18)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,15,16,19)(10,20,21,14)(11,24,17,13)(12,23,22,18)(25,29)(26,32)(27,33)(28,37)(30,34)(31,36)(35,40)(38,39)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=gh,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, 9694656);