Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,8)(9,10,14,18,13,17,15,19)(11,12,21,22,20,23,16,24);(1,2)(3,5)(4,6)(7,8)(9,11,15,21,13,20,14,16)(10,12,19,22,17,23,18,24);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,22)(17,21)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,e^b=ei,f^b=fh,d^c=di,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313861, #72313866, #72313884, … |
| maximal normal subgroup of | #72313861, #72313866, #72313884, … |
| maximal quotient of | #72314119, #72314249, C2×#106080 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13276);