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 | #38944 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,15)(13,23)(16,22)(17,24)(26,31)(27,29)(33,36)(35,37);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(10,15)(12,21)(16,22)(18,19)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,40)(37,39);(1,3,5,4)(2,7,6,8)(9,11,14,20)(10,16,15,22)(12,19,21,18)(13,17,23,24)(25,27,30,29)(26,32,31,34)(28,35,38,37)(33,39,36,40);(9,12)(10,17)(11,19)(13,22)(14,21)(15,24)(16,23)(18,20)(25,28,30,38)(26,33,31,36)(27,35,29,37)(32,39,34,40);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,22,21,16)(25,29,30,27)(26,34,31,32)(28,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|b^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549757);