Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479580 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17216 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
895 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 895 conjugacy classes of subgroups — too many to draw. Only its 160 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 — 4 generators on 32 points:
(1,2)(3,12)(4,16)(5,9)(6,10)(7,14)(8,17)(11,19)(13,23)(15,22)(18,25)(20,26)(21,28)(24,29)(27,30)(31,32);(1,3)(2,7,10,12)(4,13)(5,15)(6,14)(8,23,16,20)(9,21,11,22)(17,26)(18,27)(19,28)(24,30,25,31)(29,32);(1,4)(2,8,10,16)(3,13)(5,18)(6,17)(7,20,12,23)(9,24,11,25)(14,26)(15,27)(19,29)(21,31,22,30)(28,32);(2,9)(5,19)(7,21)(8,24)(10,11)(12,22)(13,26)(15,28)(16,25)(18,29)(20,30)(23,31)Polycyclic presentation:
2^9|b^2=g,c^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,f^c=fi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72339012, #72339027, #72339032 |
| maximal normal subgroup of | #72339012, #72339027, #72339032 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386776);