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 | |
4457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,21,24,23)(25,26,31,33)(27,37,38,32)(28,30,36,34)(29,40,39,35);(1,2)(3,7)(4,8)(5,6)(25,27)(26,32)(28,39)(29,36)(30,40)(31,38)(33,37)(34,35);(1,3,5,4)(2,7,6,8)(10,15)(11,18)(17,20)(19,24)(26,33)(27,38)(28,36)(35,40);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,28,31,36)(26,34,33,30)(27,39,38,29)(32,35,37,40);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,29,31,39)(26,35,33,40)(27,36,38,28)(30,32,34,37);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,30,31,34)(26,36,33,28)(27,40,38,35)(29,37,39,32)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7812673);