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 | #54388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(1,2,5,6)(3,7,8,4)(10,12)(15,17)(16,20)(22,23)(26,28)(27,33)(32,36)(35,39);(2,6)(4,7)(13,21)(16,20)(18,24)(22,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3,5,8)(2,7,6,4)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8265469);