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 | |
4318 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)(12,20)(15,21)(18,24)(22,23)(25,26)(27,34)(28,29)(30,33)(31,32)(35,40)(36,38)(37,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,21)(13,17)(14,19)(15,20)(18,22)(23,24)(26,32)(28,37)(33,34)(35,38);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(25,27,31,30)(26,33,32,34)(28,35,37,38)(29,36,39,40);(9,12)(10,15)(11,18)(13,20)(14,22)(16,23)(17,21)(19,24)(25,28)(26,29)(27,35)(30,38)(31,37)(32,39)(33,36)(34,40);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29)(26,28)(27,36)(30,40)(31,39)(32,37)(33,35)(34,38);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,30,31,27)(26,33,32,34)(28,38,37,35)(29,36,39,40)Polycyclic presentation:
2^9|c^2=g,e^2=gh,f^2=h,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, 7565441);