Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144627 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 663 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 3 generators on 32 points:
(1,2)(3,5)(4,14,18,30)(6,22)(7,10)(8,17,24,32)(9,23)(11,26,15,31)(12,28)(13,21)(16,29,19,27)(20,25);(1,3,5,2)(4,15,16,8)(6,9,20,12)(7,13,21,10)(11,19,24,18)(14,17,29,31)(22,28,25,23)(26,30,32,27);(1,4)(2,8)(3,11)(5,19)(6,17)(7,18)(9,14)(10,24)(12,27)(13,15)(16,21)(20,26)(22,30)(23,32)(25,29)(28,31)Polycyclic presentation:
2^9|a^2=di,b^2=d,b^a=bd,c^a=ce,e^a=eg,f^a=fghi,c^b=cf,e^b=eh,f^b=fg,d^c=dg,h^c=hi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72263134, #72263160, #72263164, … |
| maximal normal subgroup of | #72263134, #72263160, #72263164, … |
| maximal quotient of | #72266620, #72266623, #72337748 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51824);