Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24);(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(1,3)(2,5)(4,8)(6,7)(11,18)(14,17)(19,24)(21,23);(10,15)(14,17)(16,22)(21,23);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235084, #72235087, #72235094, … |
| maximal normal subgroup of | #72235084, #72235087, #72235094, … |
| maximal quotient of | #72236583, #72236776, #72257554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281772);