Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8);(1,3,4,7)(2,6,5,8);(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^9|a^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf176 conjugacy classes — view the full character table.
| maximal subgroup of | #72313295, #72313298, #72313320, … |
| maximal normal subgroup of | #72313295, #72313298, #72313320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529921);