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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(13,21)(17,22)(18,24)(20,23)(25,26,30,33)(27,28,31,36)(29,37,38,32)(34,40,39,35);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,30,31)(26,28,33,36)(29,34,38,39)(32,35,37,40);(1,2,5,4)(3,6,8,7)(10,15)(12,16)(17,22)(20,23)(25,26)(27,28)(29,32)(30,33)(31,36)(34,35)(37,38)(39,40);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,39)(34,37)(38,40);(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,37)(36,40);(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622806);