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 | #50768 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26)(27,34)(28,32)(29,31)(30,37)(33,38)(35,39)(36,40);(1,2)(3,6)(4,7)(5,8)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,3,5,4)(2,6,8,7)(10,15)(11,17)(16,22)(18,23)(26,29)(27,28)(33,37)(35,36);(1,4,5,3)(2,7,8,6)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7626297);