Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1964 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10960 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,7,13,5,11)(3,14,17,9,18,28,16,29)(4,10,20,24,21,30,22,31)(8,23,26,19,27,32,25,15);(1,3)(2,8,13,27)(4,15,21,19)(5,17,6,16)(7,18)(9,24,29,31)(10,28)(11,25)(12,26)(14,30)(20,32)(22,23);(1,3)(2,9)(4,19,21,15)(5,16,6,17)(7,18)(8,24)(10,26,30,25)(11,28,12,14)(13,29)(20,32)(22,23)(27,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi,f^d=fi,g^d=gi,f^e=fi,g^e=gi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280582, #72281046, #72282782, … |
| maximal normal subgroup of | #72280582, #72281046, #72282782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54715);