Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,6,4,7)(11,18)(12,13)(15,19)(17,22)(20,21)(23,24);(2,6)(3,4)(7,8)(9,10)(11,19)(12,17)(13,22)(14,16)(15,21)(18,23)(20,24);(2,7)(3,4)(6,8)(9,11)(10,15)(12,21)(13,18)(14,20)(16,23)(17,24)(19,22);(10,16)(15,23)(17,22)(19,24)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72316816, #72316965, #72316966 |
| maximal normal subgroup of | #72316816, #72316965, #72316966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141086);