Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7414 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1345 |
| commutator | C2×M16 | C42 |
| Frattini, cosocle | C2×#423 | C22 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C2×#423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
296 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,14)(11,17)(13,16)(15,19)(18,20);(6,10,12,18,14,19,13,11)(7,15)(8,16)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,e^2=h,f^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,h^a=hi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,g^c=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72209821, #72209823 |
| maximal normal subgroup of | #72209821 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1527);