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 | #51940 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4614 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)(11,13)(12,19)(14,15)(16,20)(17,23)(18,24)(21,22)(25,26,30,32)(27,29,35,31)(28,33,37,40)(34,38,36,39);(2,6)(3,7)(10,15)(11,17)(12,20)(22,24)(25,27)(26,31)(28,36)(29,32)(30,35)(33,38)(34,37)(39,40);(1,3)(2,4)(5,7)(6,8)(11,17)(13,23)(18,21)(22,24)(26,32)(28,37)(29,31)(34,36);(1,4)(2,3)(5,8)(6,7)(9,11)(10,13)(12,21)(14,17)(15,23)(16,24)(18,20)(19,22)(25,28)(26,33)(27,34)(29,38)(30,37)(31,39)(32,40)(35,36);(9,12)(10,16)(11,18)(13,22)(14,20)(15,19)(17,21)(23,24)(25,27)(26,31)(28,36)(29,32)(30,35)(33,38)(34,37)(39,40);(9,13)(10,11)(12,22)(14,23)(15,17)(16,18)(19,21)(20,24)(25,29,30,31)(26,27,32,35)(28,38,37,39)(33,34,40,36)Polycyclic presentation:
2^9|a^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841537);