Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,25)(17,19)(18,23)(20,26)(22,27)(24,28);(13,15)(14,18)(16,22)(17,24)(19,27)(20,28)(21,25)(23,26);(6,9)(10,12)(14,19)(18,27)(20,25)(21,28);(14,19)(15,22)(16,26)(18,21)(23,24)(27,28);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,d^b=df,f^b=fi,f^d=fi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72319044, #72319057, #72319072, … |
| maximal normal subgroup of | #72319044, #72319057, #72319072, … |
| maximal quotient of | #72322975 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288069);