Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134690 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#106 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #106 |
| 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 | |
352 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 114 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)(5,6,9,7,8,10,12,11)(13,14,15,20,18,22,17,27)(16,24,23,28,26,19,25,21);(5,7,9,10,8,11,12,6)(14,19,22,24)(15,17)(16,25)(20,28,27,21)(23,26);(2,4)(6,10)(7,11)(13,15,18,17)(14,20,22,27)(16,23,26,25)(19,21,24,28)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=i,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,f^b=fi,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224980, #72225053, #72225063, … |
| maximal normal subgroup of | #72224980, #72225053, #72225063, … |
| maximal quotient of | #72224971, #72224972, #72224976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43042);