Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1504 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(10,14)(13,16)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(1,4,5,3)(2,7,6,8)(18,23)(22,30)(24,31)(29,32)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72308782, #72309015 |
| maximal normal subgroup of | #72308782, #72309015 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7435497);