Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499849 |
| 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 | #45977 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1274 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 136 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)(10,14)(16,21)(17,24)(19,22);(2,5)(6,8)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19);(1,3,4,7)(2,6,5,8)(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17);(2,5)(6,8)Polycyclic presentation:
2^9|b^2=i,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72256806, #72256810, #72256811, … |
| maximal normal subgroup of | #72256806, #72256810, #72256811, … |
| maximal quotient of | #72317745, #72318316 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407379);