Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138552 |
| 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 | #532 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 342 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10)(11,21,24,36)(12,17,27,35)(13,28)(14,20)(15,33)(16,23,34,29)(18,22)(19,32)(25,26,39,30)(31,40)(37,38);(2,6)(3,4)(7,8)(9,11,22,35,32,39,13,29,14,24,38,17,15,25,31,23)(10,16,28,26,33,12,18,36,20,34,40,30,19,27,37,21);(2,6)(3,4)(7,8)(9,12)(10,17)(11,18)(13,30)(14,27)(15,16)(19,29)(20,35)(21,38)(22,36)(23,33)(24,37)(25,28)(26,31)(32,34)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45780);