Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6451863 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2404 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4049 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)(3,7)(4,8)(5,9)(6,10)(11,22)(12,20)(13,21)(14,19)(15,23)(16,24)(17,25)(18,26)(27,31)(28,30)(29,32);(1,3)(2,7)(4,14)(5,17)(6,13)(8,22)(9,25)(10,21)(11,16)(12,18)(15,28)(19,24)(20,26)(23,31)(27,29)(30,32);(1,4)(2,8)(3,11,13,14)(5,15)(6,16)(7,19,21,22)(9,23)(10,24)(12,27,17,28)(18,29)(20,30,25,31)(26,32);(1,5)(2,9)(3,12,13,17)(4,15)(6,18)(7,20,21,25)(8,23)(10,26)(11,27,14,28)(16,29)(19,30,22,31)(24,32);(2,10)(7,21)(8,24)(9,26)(19,22)(20,25)(23,32)(30,31)Polycyclic presentation:
2^9|c^2=g,d^2=g,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,g^b=gi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72234693, #72234953, #72237743, … |
| maximal normal subgroup of | #72234693, #72234953, #72237743, … |
| maximal quotient of | #72234587, #72234644, #72234955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359059);