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 | #51778 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(26,30)(27,33)(28,36)(39,40);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,26,29,30)(27,34,33,31)(28,32,36,38)(35,40,37,39);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,27,29,33)(26,31,30,34)(28,37,36,35)(32,40,38,39);(12,21)(15,20)(18,24)(22,23)(26,30)(31,34)(32,38)(39,40);(10,16)(14,17)(15,20)(22,23)(25,28,29,36)(26,32,30,38)(27,35,33,37)(31,39,34,40);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,e^2=i,f^2=gh,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, 7615058);