Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267419 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23851 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1258 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)(3,7,8,5)(4,6)(11,18)(15,23)(19,24)(20,21);(1,3)(2,5)(4,8)(6,7)(9,10,13,17)(11,19,18,24)(12,16,14,22)(15,20,23,21);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,15,17,23)(12,21,14,20)(16,24,22,19);(1,3)(2,5)(4,8)(6,7)(9,12,13,14)(10,16,17,22)(11,20,18,21)(15,19,23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fh53 conjugacy classes — view the full character table.
| maximal subgroup of | #72268335, #72268347, #72268369, … |
| maximal normal subgroup of | #72268335, #72268347, #72268369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174659);