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 | #42606 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,18)(13,21)(14,20)(15,17)(19,24)(22,23)(25,26,31,32)(27,33,28,37)(29,34,40,30)(35,36,38,39);(1,2,5,4)(3,7,8,6)(9,11,15,14)(10,16,17,20)(12,21,22,19)(13,23,24,18)(26,32)(29,40)(33,37)(35,38);(2,4)(6,7)(10,17)(11,14)(18,23)(19,21)(25,27,31,28)(26,33,32,37)(29,35,40,38)(30,39,34,36);(9,12)(10,18)(11,19)(13,20)(14,21)(15,22)(16,24)(17,23)(25,28,31,27)(26,33,32,37)(29,38,40,35)(30,39,34,36);(1,3,5,8)(2,6,4,7)(9,13)(10,19)(11,18)(12,20)(14,23)(15,24)(16,22)(17,21)(25,29)(26,34)(27,35)(28,38)(30,32)(31,40)(33,36)(37,39);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,20,17,16)(12,21,22,19)(13,23,24,18)(25,30,31,34)(26,29,32,40)(27,36,28,39)(33,38,37,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161566);