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 | |
2897 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)(11,16)(12,17)(13,15)(14,19)(18,21)(20,23)(22,24);(1,2,3,4)(9,11)(10,14)(12,19)(13,20)(15,22)(16,17)(18,23)(21,24);(10,15)(13,21)(14,22)(16,23)(17,18)(20,24);(5,7,8,6)(9,11)(10,16)(12,19)(13,20)(14,17)(15,23)(18,22)(21,24);(11,19)(14,16)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,g^c=gi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72284467, #72284589, #72284766, … |
| maximal normal subgroup of | #72284467, #72284589, #72284766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280023);