Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267371 |
| 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 | #23848 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(10,13)(14,16);(1,3)(2,5)(4,8)(6,7)(10,14)(12,15)(13,16);(1,2)(3,5)(4,6)(7,8)(9,10,11,13)(12,16,15,14);(1,3)(2,5)(4,8)(6,7)Polycyclic presentation:
2^9|a^2=fi,c^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,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72268289, #72268297, #72268322, … |
| maximal normal subgroup of | #72268289, #72268297, #72268322, … |
| maximal quotient of | #72313263, #72313400, #72313483 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174587);