Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #263291 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #18178 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
994 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 146 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)(4,7)(6,8)(10,14)(11,17)(12,21)(16,23)(19,20)(22,24);(3,8)(5,6)(9,10,12,14,13,16,21,23)(11,18,19,24,20,15,17,22);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,22)(16,18)(17,21)(23,24);(9,10,12,14,13,16,21,23)(11,15,19,22,20,18,17,24)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72305440, #72305442, #72305458, … |
| maximal normal subgroup of | #72305440, #72305442, #72305458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 170487);