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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4481 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,15,19,13)(12,20,22,16)(18,21,24,23)(25,26)(27,28)(29,33)(30,34)(31,32)(35,36)(37,39)(38,40);(1,3,5,8)(2,6,4,7)(25,27,30,32)(26,31,34,28)(29,35,38,39)(33,37,40,36);(9,11,14,19)(10,13,17,15)(12,18,22,24)(16,21,20,23)(25,26,30,34)(27,28,32,31)(29,33,38,40)(35,36,39,37);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,39)(38,40);(9,12,14,22)(10,16,17,20)(11,18,19,24)(13,21,15,23)(25,28,30,31)(26,32,34,27)(29,37,38,36)(33,35,40,39);(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24)(25,29,30,38)(26,33,34,40)(27,35,32,39)(28,36,31,37)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7777648);