Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152666 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13082 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #23872 | C2 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
618 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 618 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,8,14,4,22)(3,15,19,27,21,32,17,31)(5,11,24,10,7,13,23,30)(9,18,28,25,29,20,26,16);(1,3,8,21)(2,9,13,27,14,29,11,31)(4,16,6,25)(5,20,7,18)(10,26,22,32,30,28,12,15)(17,24,19,23);(1,4)(2,10)(3,16)(5,24)(6,8)(7,23)(9,26)(11,12)(13,22)(14,30)(15,31)(17,18)(19,20)(21,25)(27,32)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,e^b=ei,f^b=fg,g^b=gh,d^c=di,e^c=ei,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72283143, #72283820, #72283987 |
| maximal normal subgroup of | #72283143, #72283820, #72283987 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59864);