Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431743 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#127 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2397 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,9,6,10,18,24)(3,11,13,19,14,23,17,7)(4,8,16,22,15,21,28,31)(12,25,27,30,26,32,29,20);(1,3)(2,7)(4,12)(5,17)(6,14)(8,20)(9,23)(10,19)(11,24)(13,18)(15,26)(16,29)(21,30)(22,32)(25,31)(27,28);(7,19)(11,23)(20,30)(25,32);(1,4,5,16,6,15,18,28)(2,8,9,22,10,21,24,31)(3,12,17,29,14,26,13,27)(7,20,23,32,19,30,11,25);(4,15)(8,21)(12,26)(16,28)(20,30)(22,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=f,d^2=fg,f^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,f^b=fi,g^b=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215972, #72216095, #72216238, … |
| maximal normal subgroup of | #72215972, #72216095, #72216238, … |
| maximal quotient of | #72226715, C2×#6432582, #72252131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339778);