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 | #45822 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,21,24,23)(26,31)(27,34)(28,36)(37,38);(1,3,5,8)(2,6,4,7)(25,26,30,31)(27,32,34,29)(28,33,36,39)(35,37,40,38);(1,2,5,4)(3,6,8,7)(10,15)(11,18)(17,20)(19,24)(25,27,30,34)(26,29,31,32)(28,35,36,40)(33,38,39,37);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,37)(36,40)(38,39);(1,4,5,2)(3,7,8,6)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28)(26,33)(27,35)(29,38)(30,36)(31,39)(32,37)(34,40);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,29,30,32)(26,27,31,34)(28,37,36,38)(33,40,39,35)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7803242);