Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112147 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1141 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1141 conjugacy classes of subgroups — too many to draw. Only its 134 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:
(2,7,11,24)(3,13)(4,17)(8,25,27,26)(9,22,12,14)(10,21,28,23)(15,29)(16,31)(18,19)(30,32);(1,2)(3,14)(4,8)(5,9)(6,12)(7,15)(10,18)(11,20)(13,24)(16,21)(17,27)(19,28)(22,29)(23,32)(25,30)(26,31);(2,8)(3,15)(4,18)(7,21)(9,10)(11,27)(12,28)(13,29)(14,26)(16,32)(17,19)(22,25)(23,24)(30,31)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,f^c=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213382, #72213906, #72219490 |
| maximal normal subgroup of | #72213382, #72213906, #72219490 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19343);