Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #466 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,22,30)(15,31,36,44)(16,37,33,42)(17,26,38,27)(18,25,19,40)(20,29,21,28)(23,39,43,32)(24,35,41,34);(5,6)(7,9)(8,11)(10,12)(13,15,22,36)(14,23,30,43)(16,38,33,17)(18,32,19,39)(20,35,21,34)(24,25,41,40)(26,31,27,44)(28,37,29,42);(2,4)(6,9)(7,12)(10,11)(13,16,17,15,21,32,19,35,22,33,38,36,20,39,18,34)(14,24,25,43,29,44,27,42,30,41,40,23,28,31,26,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=d,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=egh,f^c=fgh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52732);