Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1942 |
| prime radical | C23 | C4×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 212 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) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,6,3,5,7,4,8)(10,16)(11,20)(12,14)(13,26)(17,30)(18,34)(19,25)(21,27)(22,36)(23,35)(24,37)(28,32)(33,40)(38,39);(1,3,6,7,5,8,4,2)(9,10)(11,17)(12,25)(13,18)(14,16)(15,19)(20,32)(21,31)(22,38)(23,28)(24,33)(26,29)(27,34)(30,36)(35,40)(37,39);(1,4,5,6)(2,3,7,8)(9,11,21,35,14,23,13,20,15,24,26,36,12,22,27,37)(10,17,31,40,16,28,18,32,19,33,29,30,25,38,34,39)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=egi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ch,d^b=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47564);