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 | #56315 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,18)(13,23)(14,15)(16,21)(19,24)(20,22)(28,36)(31,38)(35,39)(37,40);(2,6)(3,7)(10,15)(12,22)(16,17)(19,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,3)(2,4)(5,7)(6,8)(9,11,14,21)(10,16,15,17)(12,19,22,23)(13,20,24,18)(25,27,30,32)(26,29,33,34)(28,35,36,39)(31,37,38,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,17)(12,19)(13,20)(14,21)(15,16)(18,24)(22,23)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,38)(34,40);(9,12,14,22)(10,18,15,20)(11,19,21,23)(13,17,24,16)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40);(9,13,14,24)(10,19,15,23)(11,20,21,18)(12,16,22,17)(25,29)(26,32)(27,33)(28,37)(30,34)(31,39)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9627734);