Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| 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 | |
1034 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,22)(14,18)(15,20)(17,23)(21,24);(5,8)(9,11,14,12)(10,15,18,24)(13,21,23,20)(16,22,19,17);(1,3)(2,5)(4,7)(6,8)(9,12,14,11)(10,16,18,19)(13,21,23,20)(15,17,24,22)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,d^c=dg,e^d=ei44 conjugacy classes — view the full character table.
| maximal subgroup of | #72250099, #72256813, #72256816, … |
| maximal normal subgroup of | #72250099, #72256813, #72256816, … |
| maximal quotient of | #72250227, #72316294 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51585);