Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | #52981 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,31)(28,32)(29,37)(30,33)(34,38)(35,39)(36,40);(1,2)(3,5)(4,6)(7,8)(10,14)(15,20)(16,21)(22,24)(25,27,30,29)(26,31,33,37)(28,35,36,34)(32,39,40,38);(1,3,4,8)(2,5,6,7)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,32)(27,34)(29,35)(30,36)(31,38)(33,40)(37,39);(2,6)(5,7)(10,14)(11,17)(16,21)(18,23)(28,36)(32,40)(34,35)(38,39);(10,14)(15,20)(16,21)(22,24)(25,29,30,27)(26,31,33,37)(28,35,36,34)(32,38,40,39);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000247);