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 | #37903 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4522 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)(9,10,14,16)(11,17,12,15)(13,18,21,23)(19,24,20,22)(25,26,30,32)(27,33,34,28)(29,31,39,40)(35,38,37,36);(1,2)(3,7)(4,8)(5,6)(9,11,14,12)(10,15,16,17)(13,20,21,19)(18,24,23,22)(25,27)(26,28)(29,37)(30,34)(31,38)(32,33)(35,39)(36,40);(1,3,5,4)(2,7,6,8)(10,16)(11,12)(18,23)(19,20)(27,34)(28,33)(35,37)(36,38);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,26)(27,28)(29,31)(30,32)(33,34)(35,36)(37,38)(39,40);(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24)(25,28)(26,27)(29,38)(30,33)(31,37)(32,34)(35,40)(36,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,29,30,39)(26,31,32,40)(27,35,34,37)(28,36,33,38)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,e^2=h,f^2=g,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, 7881658);