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 | #35072 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(25,26,31,34)(27,29,28,32)(30,33,39,38)(35,37,36,40);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,29)(28,32)(30,38)(31,34)(33,39)(35,40)(36,37);(10,14)(11,17)(16,19)(18,23)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29,31,32)(26,27,34,28)(30,37,39,40)(33,35,38,36);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,39)(26,33,34,38)(27,35,28,36)(29,37,32,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644215);