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 | #59947 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4460 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)(9,10,15,18)(11,16,19,24)(12,22,21,14)(13,23,20,17)(27,32)(28,31)(29,37)(33,40);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,13)(14,17)(15,19)(18,24)(20,21)(22,23)(25,26,30,34)(27,28,32,31)(29,33,37,40)(35,36,38,39);(1,3)(2,4)(5,8)(6,7)(11,19)(12,21)(14,22)(16,24)(25,27,30,32)(26,31,34,28)(29,38,37,35)(33,36,40,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,14)(11,20)(13,19)(15,21)(16,23)(17,24)(18,22)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,40)(37,38);(9,13,15,20)(10,17,18,23)(11,21,19,12)(14,16,22,24)(25,28,30,31)(26,32,34,27)(29,36,37,39)(33,38,40,35);(9,14)(10,12)(11,17)(13,16)(15,22)(18,21)(19,23)(20,24)(25,29,30,37)(26,33,34,40)(27,35,32,38)(28,36,31,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,e^2=gi,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, 8842039);