Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,28)(31,32);(10,14)(13,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,3,4,7)(2,6,5,8)(19,26)(22,30)(25,27)(29,31);(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^9|b^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,h^b=hi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239646, #72313476, #72313494, … |
| maximal normal subgroup of | #72239646, #72313476, #72313494, … |
| maximal quotient of | #72239662, #72239673, #72319937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339676);