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 | #52669 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4218 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,35)(27,28,38,32)(29,33,30,34)(36,39,37,40);(1,2)(3,4)(5,7)(6,8)(11,17)(15,21)(18,23)(22,24)(25,27,31,38)(26,32,35,28)(29,36,30,37)(33,40,34,39);(1,3)(2,6)(4,7)(5,8)(10,14)(15,21)(16,19)(22,24)(25,27)(26,28)(29,36)(30,37)(31,38)(32,35)(33,39)(34,40);(1,4,5,6)(2,3,7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,27)(29,39)(30,40)(31,32)(33,36)(34,37)(35,38);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,30)(26,33,35,34)(27,36,38,37)(28,39,32,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,29)(26,34,35,33)(27,37,38,36)(28,40,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9405713);