Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2144 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)(11,14)(12,16)(13,15)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,32)(30,31);(9,11,13,12)(10,14,15,16)(18,23)(20,21)(24,31)(25,26)(27,32)(28,30);(2,5)(6,8)(10,15)(11,12)(17,19)(18,24)(20,30)(21,28)(22,29)(23,31)(25,32)(26,27);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(17,20,22,21)(18,25,26,23)(19,28,29,30)(24,31,27,32);(11,12)(14,16)(19,29)(24,27)(28,30)(31,32)Polycyclic presentation:
2^9|b^2=h,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72282842, #72283101, #72283180, … |
| maximal normal subgroup of | #72282842, #72283101, #72283180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6519008);