Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 100 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)(13,14,18,23,19,24,15,25)(16,21,27,28,26,20,17,22);(2,4)(5,7,9,10,8,11,12,6)(14,20,24,21)(15,18)(16,17)(22,23,28,25)(26,27);(6,10)(7,11)(13,15)(14,21)(17,27)(18,19)(20,24)(22,23)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ch,d^b=dg,e^b=egi,f^b=fi,g^b=gi,d^c=dgi,e^c=eg,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225053, #72225063, #72225064, … |
| maximal normal subgroup of | #72225053, #72225063, #72225064, … |
| maximal quotient of | #72225048, #72225059, #72225069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43128);