Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,28)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,14)(10,12,15,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(17,19)(18,22)(20,23)(21,24)(25,26)(27,28)(29,31)(30,32);(1,4,5,6)(2,3,7,8)(9,10)(11,12)(13,15)(14,16)(17,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,32)(29,31);(1,3)(2,4)(5,8)(6,7)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,23,24,22)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=fghi,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72303423, #72303502, #72303510, … |
| maximal normal subgroup of | #72303423, #72303502, #72303510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1223759);