Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235462 |
| 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 | C2×C42 | C24 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(1,2,4,6)(3,8,7,5)(14,23)(17,24);(3,7)(5,8)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(11,18)(14,23)(17,24)(19,21)Polycyclic presentation:
2^9|b^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72280491, #72280583, #72305969, … |
| maximal normal subgroup of | #72280491, #72280583, #72305969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142660);