Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
3122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,18,20,14)(12,15)(13,16)(17,22)(19,24,21,23);(2,4)(5,6,7,8)(9,11)(10,14)(12,21)(13,20)(15,24)(16,18)(17,23)(19,22);(10,15)(12,22)(14,23)(16,17)(18,24)(19,21)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72235935, #72236744, #72236752, … |
| maximal normal subgroup of | #72235935, #72236744, #72236752, … |
| maximal quotient of | #72235961, #72239023, #72281469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30756);