Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129554 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
486 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,10,11,12)(13,14,16,24,20,25,28,22,18,23,27,21,19,15,17,26);(5,6,7,10,9,11,8,12)(13,15,20,23,18,25,19,14)(16,21,28,22,27,24,17,26);(2,4)(5,7,9,8)(6,10,11,12)(13,16,20,28,18,27,19,17)(14,21,25,26,23,24,15,22)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=gi,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,d^b=dg,f^b=fg,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318709, #72318720, #72318729, … |
| maximal normal subgroup of | #72318709, #72318720, #72318729, … |
| maximal quotient of | #72320673, #72322438, #72322507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43838);