Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #2104⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,13,15,10)(8,16,18,11)(9,12)(14,19)(17,20);(5,7)(6,10)(8,17,18,14)(9,15)(11,19,16,20)(12,13);(1,3,4,2)(5,8)(6,11)(7,14)(9,18)(10,19)(12,16)(13,20)(15,17);(5,6)(7,13,15,10)(8,11)(9,12)(14,20,17,19)(16,18)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=egh,d^2=fg,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298378);