Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124934 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#331 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1100 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 3 generators on 20 points:
(1,2)(3,4)(5,6,7,12,15,10,14,18,9,13,17,20,8,11,16,19);(6,10)(7,14)(8,15)(11,13)(12,20)(16,17);(2,4)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214877, #72214886, #72219586, … |
| maximal normal subgroup of | #72214877, #72214886, #72219586, … |
| maximal quotient of | #72219627 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37182);