Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143214 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28467 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
549 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 549 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) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6)(7,9)(8,11)(10,12)(13,14,19,25)(15,26,31,37)(16,21,35,42)(17,24,18,23)(20,29,40,28)(22,34,41,33)(27,43,30,36)(32,38,44,39);(1,2)(3,4)(6,9)(7,12)(10,11)(13,15)(14,20)(16,32)(17,27)(18,30)(19,31)(21,26)(22,36)(23,39)(24,38)(25,40)(28,33)(29,34)(35,44)(37,42)(41,43);(2,4)(14,21)(15,27)(16,33)(17,18)(20,36)(22,24)(23,41)(25,42)(26,39)(28,29)(30,31)(34,35)(37,38)(40,43)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50435);