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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,15)(12,16)(18,21)(20,23)(25,26,30,34)(27,35,36,31)(28,32,37,39)(29,33,38,40);(1,3,5,8)(2,6,7,4)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(27,36)(28,37)(31,35)(32,39);(1,3)(2,4)(5,8)(6,7)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,27)(26,31)(28,33)(29,32)(30,36)(34,35)(37,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,28,30,37)(26,32,34,39)(27,33,36,40)(29,35,38,31);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19);(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29)(26,33)(27,32)(28,31)(30,38)(34,40)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=ghi,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8364145);