Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
620 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)(6,10,12,18)(7,14)(8,17)(11,16,13,19)(15,20);(7,15)(10,18)(14,20)(16,19);(5,6,8,11,9,12,17,13)(7,16,14,10,15,19,20,18);(1,3,4,2)(6,11,12,13)(7,15)(10,16,18,19)(14,20)Polycyclic presentation:
2^9|a^2=g,c^2=eh,d^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#33688, #72258431, #72265415, … |
| maximal normal subgroup of | C4×#33688, #72265415, #72307033, … |
| maximal quotient of | #72242481, #72258431 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403149);