Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419792 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1324 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)(3,4)(5,6,9,15,10,13,20,12)(7,14,16,8,17,11,19,18);(5,7,9,16,10,17,20,19)(6,8,12,14,13,18,15,11);(5,8,9,14,10,18,20,11)(6,7,15,19,13,17,12,16);(2,4)(5,7)(6,11)(8,15)(9,19)(10,17)(12,18)(13,14)(16,20)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72238735 |
| maximal normal subgroup of | #72238735 |
| maximal quotient of | #72238747 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341275);