Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#1721 | C2 |
| 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 | |
517 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 517 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)(5,6,10,15,9,18,20,17,12,13,8,14,11,7,16,19);(1,2,3,4)(5,7,12,18)(6,9,13,11)(8,15,10,14)(16,19,20,17);(6,13)(8,16)(9,11)(10,20)(14,17)(15,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=ghi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214886, #72217755, #72219587, … |
| maximal normal subgroup of | #72214886, #72217755, #72219587, … |
| maximal quotient of | #72215015, #72215017, #72215020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44752);