Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C23 | #442 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,5,7,4,3,6,8)(9,10,15,21,16,22,39,56,17,23,14,40,25,45,13,20)(11,24,29,49,30,55,27,57,31,60,28,18,43,47,41,48)(12,32,35,58,36,67,34,19,37,52,38,53,44,51,42,54)(26,61,63,69,33,46,65,59,64,50,68,72,66,71,62,70);(1,3,5,8,4,2,6,7)(9,11,25,30,17,31,16,43)(10,18,45,48,23,49,22,57)(12,33,36,26,37,66,44,64)(13,28,14,41,39,29,15,27)(19,50,53,46,54,61,58,71)(20,55,40,60,56,47,21,24)(32,59,67,69,52,70,51,72)(34,65,38,63,42,62,35,68);(2,3)(7,8)(9,12,17,37)(10,19,23,54)(11,26,31,64)(13,38,39,35)(14,42,15,34)(16,44,25,36)(18,46,49,71)(20,32,56,52)(21,51,40,67)(22,58,45,53)(24,59,60,70)(27,63,41,68)(28,62,29,65)(30,66,43,33)(47,72,55,69)(48,61,57,50)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43922);