Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150429 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
377 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,15,20,12,27)(11,21,24,16,26,33,40,34)(14,19,29,35,31,37,30,17)(22,39,23,32,25,36,28,38);(2,6)(3,4)(7,8)(9,11)(10,16)(12,28)(13,23)(14,22)(15,26)(17,21)(18,36)(19,38)(20,34)(24,30)(25,31)(27,39)(29,40)(32,37)(33,35);(1,3,5,4)(2,7,8,6)(11,22)(14,31)(16,32)(17,35)(19,37)(21,39)(23,24)(25,26)(28,40)(29,30)(33,36)(34,38)Polycyclic presentation:
2^9|a^2=e,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cfhi,d^b=dg,e^b=efgi,f^b=fi,d^c=dhi,f^c=fi,e^d=ehi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57691);