Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(10,15)(11,19,21,22)(16,20,23,24)(17,18);(2,4)(9,10)(11,20,21,24)(12,15)(13,17)(14,18)(16,22,23,19);(6,8)(9,11)(10,16)(12,21)(13,22)(14,19)(15,23)(17,20)(18,24);(2,4)(10,15)(16,23)(17,18)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239233, #72264635, #72264644, … |
| maximal normal subgroup of | #72239233, #72264635, #72264644, … |
| maximal quotient of | #72239729, #72280051, #72280080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271589);