Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,17)(12,18)(13,15)(14,22)(16,19)(20,24)(21,23);(1,2,3,4)(10,15)(16,23)(17,24)(18,22);(9,11)(10,16)(12,19)(13,20)(14,21)(15,23)(17,22)(18,24);(5,7,8,6)(9,11)(10,17)(12,21)(13,20)(14,19)(15,24)(16,22)(18,23);(9,12,13,14)(10,18,15,22)(11,19,20,21)(16,24,23,17)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,g^c=gi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72284104, #72284766, #72284895 |
| maximal normal subgroup of | #72284104, #72284766, #72284895 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280134);