Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234365 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2510 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 2510 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,17)(16,20)(18,24)(19,23)(21,22);(1,3)(2,5)(4,7)(6,8)(10,14)(12,20)(15,23)(16,17)(19,21)(22,24);(5,8)(9,11)(10,15)(12,21)(13,18)(14,22)(16,23)(17,24)(19,20);(11,18)(15,24)(19,21)(22,23)Polycyclic presentation:
2^9|e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=dh,e^c=eh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72257614, #72257649, #72257657, … |
| maximal normal subgroup of | #72257614, #72257649, #72257657, … |
| maximal quotient of | #72264345, #72264405 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 141561);