Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12298 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1613 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8403 |
| prime radical | #60741 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
804 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,5,10,6,11,22,29,7,9,4,17,19,27,20,32)(3,12,15,30,16,28,21,8,14,25,13,26,23,31,18,24);(1,3)(2,8)(4,18)(5,21)(6,23)(7,14)(9,24)(10,28)(11,30)(12,29)(13,20)(15,22)(16,19)(17,31)(25,32)(26,27);(2,9)(8,24)(11,27)(26,30)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,f^c=fg,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72217920, #72219578, #72224554, … |
| maximal normal subgroup of | #72217920, #72219578, #72224554, … |
| maximal quotient of | #72216373, #72216380, #72216384 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38510);