Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #45769 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,18,20,16)(12,17,13,22)(19,24,21,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(2,6)(7,8)(10,15)(16,18)(17,22)(23,24)(25,27,29,34)(26,30,32,38)(28,35,36,33)(31,39,40,37);(9,11)(10,16)(12,21)(13,19)(14,20)(15,18)(17,24)(22,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,15,22)(11,19,20,21)(16,23,18,24)(26,32)(28,36)(30,38)(33,35);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,15,22)(11,19,20,21)(16,24,18,23)(28,36)(31,40)(33,35)(37,39);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,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, 7609719);