Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138556 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D4 | SD16 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10,13,18,15,19,14,30)(11,20,22,37,23,36,29,38)(12,24,26,32,28,40,27,17)(16,31,33,21,35,39,34,25);(2,6)(3,4)(7,8)(9,11,12,25,14,22,27,31,15,23,28,21,13,29,26,39)(10,16,32,20,18,34,40,38,19,35,17,36,30,33,24,37);(2,6)(3,4)(7,8)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=ehi,f^c=fhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309443, #72309457, #72309458, … |
| maximal normal subgroup of | #72309443, #72309457, #72309458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45781);