Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,11,13,16,21,18)(12,15,20,17,19,22,24,23);(2,4)(9,11,14,16,13,18,21,10)(12,17,20,22,19,23,24,15);(6,8);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24);(12,19)(15,22)(17,23)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,h^2=i,b^a=bf,c^a=cg,e^d=ei140 conjugacy classes — view the full character table.
| maximal subgroup of | #72316526, #72316527, #72316530, … |
| maximal normal subgroup of | #72316526, #72316527, #72316530, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278538);