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 | #38034 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4415 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)(9,10,16,20)(11,17,22,12)(13,24,21,18)(14,19,23,15)(26,31)(27,33)(34,39)(35,40);(1,2,5,6)(3,7,4,8)(9,11)(10,17)(12,20)(13,19)(14,18)(15,21)(16,22)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,23)(14,21)(15,24)(16,17)(18,19)(20,22)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(9,13,16,21)(10,18,20,24)(11,19,22,15)(12,23,17,14);(1,4,5,3)(2,8,6,7)(9,14,16,23)(10,19,20,15)(11,18,22,24)(12,21,17,13)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(9,15,16,19)(10,14,20,23)(11,21,22,13)(12,24,17,18)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=i,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565261);