Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1898 |
| 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×#568 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8)(3,11,15,30)(5,9,12,23)(6,10,19,17)(7,22,25,29)(13,32,16,28)(14,26,31,24)(18,27,21,20);(1,2)(3,7)(4,17,19,8)(5,20,21,9)(6,10)(11,29,28,14)(12,23)(13,24,15,25)(16,26)(18,27)(22,32)(30,31);(1,3,12,31,6,16,18,22)(2,7,23,30,10,26,27,32)(4,15,21,29,19,13,5,14)(8,24,20,11,17,25,9,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,d^b=dhi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54718);