Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511998 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,9)(7,15,17,12)(8,10)(13,14)(16,18,19,20);(5,6,8,9,11,14,10,13)(7,12,16,18,17,15,19,20);(1,2,3,4);(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,20)(14,15)Polycyclic presentation:
2^9|a^2=eh,b^2=e,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^d=ghi,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#512418, #72226767 |
| maximal normal subgroup of | C2×#512418, #72226767 |
| maximal quotient of | C2×#512418 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419614);