Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6348447 |
| 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 | C42 | C25 |
| prime radical | C2×#3543 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3192 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)(3,7)(4,6)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(20,29)(23,28)(25,32)(27,30);(1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16)(17,19)(18,22)(20,30)(21,26)(23,32)(24,31)(25,28)(27,29);(3,10)(7,14)(11,16)(13,15);(2,8)(6,9)(7,14)(13,15)Polycyclic presentation:
2^9|a^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284648, #72285130, #72286735, … |
| maximal normal subgroup of | #72284648, #72285130, #72286735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255651);