Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150441 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #391 | C23 |
| prime radical | C23 | C2×#99 |
| 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 | |
272 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(2,6,7,8)(9,10,13,18,15,20,12,26)(11,21,24,33,25,38,40,16)(14,19,27,34,30,36,29,17)(22,37,23,32,39,35,28,31);(1,2)(3,6)(4,8)(5,7)(9,11,14,22,15,25,30,39)(10,16,19,32,20,33,36,31)(12,23,27,24,13,28,29,40)(17,21,26,37,34,38,18,35);(1,3,5,4)(2,6,7,8)(11,22)(14,30)(16,31)(17,34)(19,36)(21,37)(23,24)(25,39)(27,29)(28,40)(32,33)(35,38)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efghi,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, 57732);