Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511775 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1264 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1264 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(11,18)(14,23)(15,20)(19,21);(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17)Polycyclic presentation:
2^9|c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,d^c=di,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72250122, #72256804, #72256810, … |
| maximal normal subgroup of | #72250122, #72256804, #72256810, … |
| maximal quotient of | #72250184 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419316);