Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499261 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3184 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23761 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
475 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 475 conjugacy classes of subgroups — too many to draw. Only its 126 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:
(3,12)(14,16)(15,30)(19,23);(1,2,7,10,4,11,6,21)(3,13,14,32,12,29,16,27)(5,18,22,28,17,31,20,9)(8,23,26,15,25,19,24,30);(1,3)(2,8)(4,12)(5,19)(6,14)(7,16)(9,27)(10,24)(11,25)(13,31)(15,22)(17,23)(18,29)(20,30)(21,26)(28,32);(3,14,12,16)(5,17)(8,24,25,26)(9,28)(13,32,29,27)(15,23,30,19)(18,31)(20,22)Polycyclic presentation:
2^9|b^2=fh,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=ghi,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#499601, #72283942, … |
| maximal normal subgroup of | #72237280, C2×#499601, #72283942, … |
| maximal quotient of | C2×#499601, #72257148, #72257165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406797);