Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356633 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62001 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11)(4,9)(5,10)(6,16)(7,8)(12,15)(13,14)(17,18,23,26)(19,21,25,22)(20,27,30,28)(24,31,32,29);(1,3,6,9,7,12,5,13)(2,4,10,11,8,14,16,15)(17,19,21,26,23,25,22,18)(20,28,29,24,30,27,31,32);(2,8)(3,12)(9,13)(10,16)(17,20,23,30)(18,24,26,32)(19,27,25,28)(21,29,22,31);(1,4,7,14)(2,9,8,13)(3,10,12,16)(5,15,6,11);(1,5,7,6)(2,10,8,16)(3,13,12,9)(4,11,14,15)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,g^c=gi,e^d=ehi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72340135, #72340136, #72340139, … |
| maximal normal subgroup of | #72340135, #72340136, #72340139, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263868);