Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249391 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #36702 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1734 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1734 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,11)(4,15)(5,8)(6,10)(7,13)(9,19)(12,27)(14,26)(16,23)(17,21)(18,25)(20,24)(22,28)(29,32)(30,31);(4,16)(7,21)(9,25)(11,27)(13,30)(15,23)(17,28)(18,19)(22,31)(26,32);(1,3,6,14)(2,7,10,21)(4,17,19,13)(5,12,20,29)(8,22,24,31)(9,26,15,11)(16,28,18,30)(23,32,25,27);(2,8)(4,18)(7,22)(10,24)(13,28)(16,19)(17,30)(21,31)Polycyclic presentation:
2^9|c^2=i,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,e^c=eh,f^c=fi,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72282964, #72283074, #72283590, … |
| maximal normal subgroup of | #72282964, #72283074, #72283590, … |
| maximal quotient of | #72227855, #72284120 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156599);