Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1155 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)(5,6,10,15,11,14,18,19)(7,12,8,17,13,9,16,20);(5,7)(6,12,14,9)(8,18)(10,16)(11,13)(15,20,19,17);(1,3,4,2)(5,8,10,7,11,16,18,13)(6,9,19,20,14,12,15,17);(5,9)(6,13)(7,14)(8,19)(10,20)(11,12)(15,16)(17,18)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,h^a=hi,c^b=cef,d^b=de,e^b=ei,g^b=gi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264573);