Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196332 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35272 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,9,13,10,12,18,8)(7,11,16,14,15,19,20,17);(1,2,4,3)(5,7)(6,11)(8,17)(9,16)(10,15)(12,19)(13,14)(18,20);(5,8,9,12,10,13,18,6)(7,14,20,19,15,17,16,11);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72340183, #72340189 |
| maximal normal subgroup of | #72340183, #72340189 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281552);