Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(10,14)(11,12)(15,16);(5,6)(7,8)(9,10)(11,14)(12,16)(13,15);(2,4)(6,8)(9,11,13,12)(10,14,15,16);(5,6)(7,8)(9,11,13,12)(10,14,15,16);(6,8)Polycyclic presentation:
2^9|c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72281323, #72281349, #72281374, … |
| maximal normal subgroup of | #72281323, #72281349, #72281374, … |
| maximal quotient of | #72305517, #72305521, #72305527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480750);