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 | #33190 | C2 |
| 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 | |
416 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 416 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) = 128 |
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)(11,21,25,39,26,36,24,38)(13,28,31,18,32,37,30,27)(17,33,35,23,22,40,34,29);(1,3,6,8,5,7,4,2)(9,11,13,29,15,24,31,40,16,26,32,23,14,25,30,33)(10,17,27,21,19,35,37,39,12,22,18,36,20,34,28,38);(1,4,5,6)(2,7,8,3)(9,12,16,10)(11,22,26,17)(13,18,32,27)(14,19,15,20)(21,29,36,23)(24,34,25,35)(28,30,37,31)(33,39,40,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,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, 45373);