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 | #44537 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(10,15)(11,17)(13,21)(19,24)(25,26,31,28)(27,34,33,30)(29,32,37,36)(35,40,39,38);(2,6)(3,8)(9,10,14,15)(11,12,17,20)(13,22,21,16)(18,24,23,19)(29,37)(32,36)(35,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,11,14,17)(10,12,15,20)(13,18,21,23)(16,19,22,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,40)(37,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28,31,26)(27,34,33,30)(29,36,37,32)(35,40,39,38);(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29,31,37)(26,32,28,36)(27,35,33,39)(30,38,34,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30,31,34)(26,33,28,27)(29,38,37,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8721186);