Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1409 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(10,14,15,16);(1,3)(2,5)(4,8)(6,7)(9,10,12,14,13,15,11,16);(1,2)(3,5)(4,6)(7,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=eh,f^b=fhi,d^c=dh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72231140, #72231150, #72231156, … |
| maximal normal subgroup of | #72231140, #72231150, #72231156, … |
| maximal quotient of | #72240353, C2×#106601, #72261693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13797);