Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,6,7,5,8,4,3)(9,10,15,20,16,12,14,19)(11,21,26,39,27,36,25,38)(13,29,32,18,22,37,31,28)(17,33,35,24,23,40,34,30);(1,3,6,8,5,7,4,2)(9,11,22,24,15,25,31,33,16,27,13,30,14,26,32,40)(10,17,18,36,19,35,29,38,12,23,28,21,20,34,37,39);(1,4,5,6)(2,7,8,3)(9,12,16,10)(11,23,27,17)(13,18,22,28)(14,19,15,20)(21,30,36,24)(25,34,26,35)(29,31,37,32)(33,39,40,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfghi,c^2=h,d^2=ghi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45374);