Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417901 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
766 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 170 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 — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,24)(22,23);(2,5)(3,7)(9,11,12,20,14,22,13,21)(10,15,17,19,18,24,16,23);(10,16,18,17)(11,20,22,21);(1,3,4,7)(2,6,5,8)(11,20,22,21)(15,23,24,19)Polycyclic presentation:
2^9|b^2=gh,c^2=fg,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72339028, #72339039, #72339054, … |
| maximal normal subgroup of | #72339028, #72339039, #72339054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361104);