Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150390 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11744 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| prime radical | #33748 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
232 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,9)(4,17,20,8)(5,22)(6,24)(7,13)(10,18,29,32)(11,16)(12,14)(15,25,19,26)(21,27,31,30)(23,28);(2,8)(3,5)(4,18)(6,14)(9,10)(11,29)(12,30)(13,17)(15,21)(16,23)(19,31)(20,32)(22,26)(24,27)(25,28);(1,3,14,23,7,16,6,5)(2,9,12,28,13,11,24,22)(4,19,31,18,20,15,21,32)(8,25,27,10,17,26,30,29)Polycyclic presentation:
2^9|a^2=e,c^2=fgi,d^2=ghi,f^2=h,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=ghi,h^a=hi,c^b=ch,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214318, #72214585, #72214589, … |
| maximal normal subgroup of | #72214318, #72214585, #72214589, … |
| maximal quotient of | #72216293 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57587);