Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357716 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,7)(8,13)(9,14)(10,15)(12,16)(19,25)(20,26)(22,28)(24,29)(27,31)(30,32);(1,2,7,15,5,10,4,6)(3,9,16,13,11,8,12,14)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,3)(2,8)(4,12)(5,11)(6,13)(7,16)(9,10)(14,15)(17,19,26,31,21,27,20,25)(18,22,29,32,23,30,24,28);(1,3)(2,9)(4,12)(5,11)(6,14)(7,16)(8,10)(13,15)(18,23)(22,30)(24,29)(28,32);(3,11)(8,9)(12,16)(13,14)Polycyclic presentation:
2^9|b^2=fh,c^2=gi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72339745, #72339782, #72339799, … |
| maximal normal subgroup of | #72339745, #72339782, #72339799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264930);