Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #240817 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1226 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)(3,5)(4,7)(6,8)(9,10)(11,12)(13,15)(14,16);(3,8)(5,6)(10,15)(11,16)(12,13);(2,5)(3,8)(6,7)(9,11,13,16,14,15,12,10);(9,12,14,13)(10,11,16,15)Polycyclic presentation:
2^9|c^2=e,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,f^c=fhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226222, #72226231, #72226270 |
| maximal normal subgroup of | #72226222, #72226231, #72226270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 148015);