Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150146 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11663 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | #346 | C23 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 230 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:
(2,7,8,23,10,21,11,25)(3,12,14,28)(4,17)(5,19)(9,13,18,30,22,29,27,24)(15,26)(16,32,31,20);(1,2)(3,13)(4,18)(5,8)(6,10)(7,20)(9,26)(11,19)(12,23)(14,29)(15,27)(16,24)(17,22)(21,32)(25,28)(30,31);(3,14)(7,21)(12,28)(13,29)(16,31)(20,32)(23,25)(24,30)Polycyclic presentation:
2^9|a^2=e,d^2=fi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=dfh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214531, #72214568, #72214585, … |
| maximal normal subgroup of | #72214531, #72214568, #72214585, … |
| maximal quotient of | #72216279, #72225024, C2×#150146 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57342);