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 | C2×D42 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4565 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(25,26)(27,29)(28,36)(30,34)(31,32)(33,39)(35,40)(37,38);(1,2,4,5)(3,6,7,8)(9,10,14,12)(11,15,18,17)(13,20,21,16)(19,23,24,22)(26,32)(27,30)(33,36)(35,38);(10,12)(11,18)(16,20)(19,24)(25,27,31,30)(26,29,32,34)(28,35,39,38)(33,37,36,40);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,28,31,39)(26,33,32,36)(27,35,30,38)(29,37,34,40);(9,12,14,10)(11,17,18,15)(13,16,21,20)(19,22,24,23)(25,29)(26,27)(28,37)(30,32)(31,34)(33,35)(36,38)(39,40);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,30,31,27)(26,34,32,29)(28,38,39,35)(33,40,36,37)Polycyclic presentation:
2^9|b^2=ghi,c^2=hi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7893540);