Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator, socle | C23 | C22×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,11,9,7,10,12)(13,14,17,15,16,19,18,20);(2,4)(13,14,18,20,16,19,17,15);(5,7,8,12,9,6,10,11)(13,15,17,19,16,20,18,14);(6,7)(11,12)(14,19)(15,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dh,d^b=di,d^c=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72214806, #72226829, #72239117 |
| maximal normal subgroup of | #72214806, #72226829, #72239117 |
| maximal quotient of | #72214267, #72224873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89128);