Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35346 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3999 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)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(17,25)(24,29)(26,30)(27,31)(28,32);(1,3)(2,7)(4,11)(5,12)(6,14)(8,20)(9,19)(10,21)(13,16)(15,26)(17,28)(18,23)(22,31)(24,30)(25,32)(27,29);(1,4)(2,8)(3,11)(5,15)(6,16)(7,18)(9,24)(10,23)(12,26)(13,14)(17,29)(19,31)(20,21)(22,25)(27,28)(30,32);(1,5)(2,9)(3,12)(4,15)(6,17)(7,19)(8,22)(10,25)(11,26)(13,27)(14,28)(16,29)(18,30)(20,31)(21,32)(23,24);(1,4)(2,8)(3,13)(5,15)(6,16)(7,20)(9,22)(10,23)(11,14)(12,27)(17,29)(18,21)(19,31)(24,25)(26,28)(30,32)Polycyclic presentation:
2^9|b^a=bf,c^a=cg,h^a=hi,c^b=ch,e^b=ehi,d^c=dh,f^c=fi,g^d=gi,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233206, #72234693, #72237815, … |
| maximal normal subgroup of | #72233206, #72234693, #72237815, … |
| maximal quotient of | #72233072, #72234418, #72234630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407072);