Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235159 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35001 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24);(2,6)(3,4)(7,8)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(2,7)(3,4)(6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(11,18)(14,21)(17,24)(20,22)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,e^c=ei,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257469, #72257476, #72257485, … |
| maximal normal subgroup of | #72257469, #72257476, #72257485, … |
| maximal quotient of | #72264357, #72264379 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142356);