Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #237283 |
| 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 | #60968 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1456 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)(3,7)(4,6)(5,8)(10,15)(11,19)(12,13)(17,18)(20,24)(21,22);(2,6)(3,4)(7,8)(9,10)(11,20)(12,18)(13,15)(14,17)(16,19)(21,23)(22,24);(1,3,5,4)(2,7,8,6)(9,11,14,22)(10,16,17,23)(12,19,13,21)(15,20,18,24);(1,3,5,4)(2,7,8,6)(10,17)(15,18)(16,23)(20,24)Polycyclic presentation:
2^9|c^2=h,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gi,e^c=ei,f^c=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72316851, #72316975 |
| maximal normal subgroup of | #72316851, #72316975 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144497);