Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1238 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 1238 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,7,10,8,11,12,9)(14,17)(19,20);(1,2,4,3)(13,14,18,19,15,17,16,20);(6,9)(7,12)(10,11)(14,19)(16,18)(17,20);(1,3,4,2)Polycyclic presentation:
2^9|a^2=f,b^2=eg,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72224776, #72224777, #72224779, … |
| maximal normal subgroup of | #72224776, #72224777, #72224779, … |
| maximal quotient of | #72226809 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 228613);