Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500887 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 158 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:
(1,2,4,6)(3,5,7,8)(10,14)(11,19,21,17)(12,18)(13,15)(20,22,23,24);(1,3)(2,5)(4,7)(6,8)(9,10,13,14,16,12,15,18)(11,17,20,22,21,19,23,24);(3,7)(5,8)(9,11)(10,17)(12,19)(13,20)(14,22)(15,23)(16,21)(18,24);(9,12,13,18,16,10,15,14)(11,19,20,24,21,17,23,22)Polycyclic presentation:
2^9|a^2=egh,b^2=e,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265494, #72265562, #72265621, … |
| maximal normal subgroup of | #72265494, #72265562, #72265621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 412924);