Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(9,10,14,16)(11,12,15,13);(5,6)(7,8)(9,11,13,10,14,15,12,16);(1,2,3,4)(6,8);(6,8);(9,12,14,13)(10,15,16,11)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226829, #72281444, D4×#3006, … |
| maximal normal subgroup of | #72226829, #72281444, D4×#3006, … |
| maximal quotient of | #72224851, #72224884, #72305852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350297);