Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244655 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,6)(3,7)(4,8)(9,10)(11,14)(12,15)(13,16);(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,14)(13,16);(4,8)(6,7)(10,14)(11,12)(15,16);(1,3,5,2)(4,6,8,7)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi,e^d=eh,f^d=fh53 conjugacy classes — view the full character table.
| maximal subgroup of | #72226259 |
| maximal normal subgroup of | #72226259 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151861);