Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(10,15)(11,19)(12,13)(16,24)(17,18)(21,22);(5,6)(7,8)(9,10,13,15,14,17,12,18)(11,20,21,24,22,23,19,16);(2,4)(6,8)(9,11)(10,16)(12,21)(13,19)(14,22)(15,23)(17,24)(18,20);(10,17)(15,18)(16,24)(20,23);(9,12,14,13)(10,18,17,15)(11,21,22,19)(16,20,24,23)Polycyclic presentation:
2^9|a^2=fg,b^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308796, #72308847, #72308882, … |
| maximal normal subgroup of | #72308796, #72308847, #72308882, … |
| maximal quotient of | #72316026, #72316033 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339339);