Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4294 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23);(2,5)(6,8)(25,26,29,32)(27,33,35,30)(28,36,38,31)(34,39,37,40);(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,27)(26,30)(28,37)(29,35)(31,40)(32,33)(34,38)(36,39);(11,18)(14,17)(19,24)(22,23)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(27,35)(30,33)(34,37)(39,40);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,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, 7614080);