Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(1,3)(2,5)(4,8)(6,7);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23);(10,16)(14,17)(15,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72339484, #72339486, #72339494, … |
| maximal normal subgroup of | #72339484, #72339486, #72339494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293418);