Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501084 |
| 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 | #24222 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1034 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 132 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:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,15,18,21)(13,20,19,16)(17,23,24,22);(1,3,5,4)(2,6,8,7)(10,14)(11,17)(13,19)(15,22)(18,24)(21,23);(1,3,5,4)(2,7,8,6)(9,11,12,18)(10,15,14,21)(13,17)(16,23)(19,24)(20,22);(1,4,5,3)(2,7,8,6)(10,14)(15,21)(16,20)(22,23)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=ef,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72341388, #72341394, #72341408, … |
| maximal normal subgroup of | #72341388, #72341394, #72341408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410778);