Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,8)(9,10,11,12)(13,14,18,15)(16,19,17,20);(1,2,3,4)(13,15,17,20,18,14,16,19);(6,8)(13,14)(15,18)(16,19)(17,20);(13,16,18,17)(14,19,15,20)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72315982, #72316033 |
| maximal normal subgroup of | #72315982, #72316033 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264996);