Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499837 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62819 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31);(4,12)(7,16)(9,11)(13,15);(3,10)(6,14)(9,11)(13,15)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268290, #72268296, #72268326 |
| maximal normal subgroup of | #72268290, #72268296, #72268326 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415875);