Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #237997 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#400 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,13,12,14,16,18,22)(11,17,19,24,20,21,23,15);(1,2)(3,5)(4,6)(7,8)(9,11,18,19,14,20,13,23)(10,15,22,17,16,24,12,21);(2,5)(3,8)(6,7)(9,12,13,16,14,22,18,10)(11,17,19,24,20,21,23,15);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265261, #72265439, #72265444, … |
| maximal normal subgroup of | #72265261, #72265439, #72265444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151287);