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 | #59394 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4398 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)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(26,28)(27,34)(33,36)(35,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,26,30,28)(27,32,34,31)(29,36,37,33)(35,38,40,39);(10,17)(12,22)(16,19)(20,23)(25,27,30,34)(26,31,28,32)(29,35,37,40)(33,38,36,39);(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,28,30,26)(27,31,34,32)(29,33,37,36)(35,39,40,38);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,29,30,37)(26,33,28,36)(27,35,34,40)(31,38,32,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,e^2=g,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, 7881163);