Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12301 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1613 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8873 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
406 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,7,14)(3,11)(4,20)(5,23,24,13)(6,12,25,31)(8,26,9,15)(10,22)(16,32)(17,27)(18,29)(19,30)(21,28);(1,3,9,17,7,19,8,18)(2,10,26,28,14,20,15,32)(4,5,21,25,22,24,16,6)(11,12,29,13,30,31,27,23);(1,4)(2,11,14,30)(3,5)(6,18)(7,22)(8,16)(9,21)(10,12,20,31)(13,32,23,28)(15,27,26,29)(17,25)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=gh,d^2=g,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fi,g^c=gi,h^c=hi,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72217920, #72219578, #72224559, … |
| maximal normal subgroup of | #72217920, #72219578, #72224559, … |
| maximal quotient of | #72214477, #72214486, #72214501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43460);