Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499855 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35346 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,9)(5,17)(6,11)(7,13)(10,20)(12,24)(14,23)(15,28)(16,27)(18,21)(19,26)(22,31)(25,29)(30,32);(1,3)(2,7)(4,12)(5,18)(6,14)(8,17)(9,25)(10,23)(11,21)(13,20)(15,31)(16,30)(19,29)(22,27)(24,28)(26,32);(7,21)(8,23)(22,25)(24,32);(1,3)(2,8)(4,12)(5,13)(6,14)(7,17)(9,24)(10,21)(11,23)(15,29)(16,30)(18,20)(19,31)(22,26)(25,28)(27,32);(1,4,6,16)(2,9,11,27)(3,12,14,30)(5,19,20,15)(7,22,21,25)(8,24,23,32)(10,28,17,26)(13,31,18,29)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,g^d=gi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233777, #72260900, #72261107, … |
| maximal normal subgroup of | #72233777, #72260900, #72261107, … |
| maximal quotient of | #72267103, #72306347, #72306354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407054);