Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5,6,7)(9,10,13,18,14,16,12,17)(11,19,22,15,20,24,21,23);(1,2,3,7,4,6,8,5)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,24)(18,19);(10,16)(12,13)(15,23)(21,22);(2,6)(5,7)(10,16)(11,20)(17,18)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102192);