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 | |
4635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,26,30,31)(27,28,32,36)(29,37,38,33)(34,40,39,35);(1,3,4,7)(2,5,6,8)(10,15)(12,21)(16,23)(18,22)(25,27,30,32)(26,28,31,36)(29,34,38,39)(33,35,37,40);(1,2)(3,5)(4,6)(7,8)(11,19)(13,24)(16,23)(18,22)(26,31)(27,32)(29,38)(35,40);(9,11)(10,16)(12,22)(13,20)(14,19)(15,23)(17,24)(18,21)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,21)(10,17,15,20)(11,18,19,22)(13,23,24,16)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,37)(38,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,18)(11,20)(12,23)(14,24)(15,22)(16,21)(17,19)(25,29)(26,33)(27,34)(28,35)(30,38)(31,37)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834147);