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 | #47838 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,17)(12,18)(13,19)(14,16)(15,21)(20,23)(22,24)(25,26)(27,28)(29,31)(30,33)(32,34)(35,36)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,17)(18,24)(22,23)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,2,5,4)(3,6,8,7)(10,16)(13,24)(15,17)(18,20)(26,29)(27,34)(30,38)(36,39);(1,2,5,4)(3,6,8,7)(9,11)(10,17)(12,22)(13,20)(14,21)(15,16)(18,24)(19,23)(25,28)(26,27)(29,34)(30,39)(31,32)(33,40)(35,37)(36,38);(9,12,14,23)(10,18,16,20)(11,19,21,22)(13,15,24,17)(25,29,31,26)(27,28,34,32)(30,37,38,33)(35,36,40,39);(1,4,5,2)(3,7,8,6)(9,13)(10,19)(11,20)(12,15)(14,24)(16,22)(17,23)(18,21)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7988004);