Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1950 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)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(19,27)(20,21)(23,30)(25,26)(28,29)(31,32);(10,14)(13,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,31)(25,30)(26,32);(1,4,5,3)(2,6,7,8)(18,24)(23,31)(25,26)(30,32);(17,20,22,21)(18,25,24,26)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,h^b=hi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281059, #72281161, #72281369, … |
| maximal normal subgroup of | #72281059, #72281161, #72281369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339859);