Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(6,8)(10,14,16,23)(11,18)(12,21)(15,22,24,17)(19,20);(1,2,3,4)(9,10,11,14,12,17,18,24,13,16,20,23,21,22,19,15);(5,6,7,8)(10,15)(11,19)(12,21)(14,22)(16,24)(17,23)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,g^b=gi,d^c=dhi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249883, #72249886, #72249925, … |
| maximal normal subgroup of | #72249883, #72249886, #72249925, … |
| maximal quotient of | #72264829, #72279865, C2×#149242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56438);