Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180872 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,15,19,14,28)(11,16,22,31,21,30,23,38)(12,24,26,17,27,32,25,33)(20,34,37,29,35,40,36,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,29)(18,31)(19,30)(24,34)(25,36)(26,37)(27,35)(28,38)(32,40)(33,39);(3,8)(5,7)(9,12,14,26,15,27,13,25)(10,17,18,24,19,33,28,32)(11,20,23,37,21,35,22,36)(16,29,31,34,30,39,38,40);(2,6)(5,7)(11,21)(16,30)(20,35)(22,23)(29,39)(31,38)(34,40)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,h^a=hi,c^b=ce,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263141);