Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12364 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
463 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 463 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(2,4)(5,6,10,14,13,15,19,17,12,16,9,20,11,7,8,18);(1,2,3,4)(5,7,12,15)(6,13,16,11)(8,17,19,18)(9,20,10,14);(5,8,12,19)(6,14,16,20)(7,17,15,18)(9,11,10,13)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=gi,e^2=ghi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fgi,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214877, #72219587, #72219588, … |
| maximal normal subgroup of | #72214877, #72219587, #72219588, … |
| maximal quotient of | #72221565, #72242506 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44786);