Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2355 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(18,20)(19,23)(25,26,28,31)(27,30,32,29);(1,2,4,6)(3,5,7,8)(9,11)(10,13)(12,16)(14,15)(19,23)(22,24)(25,27)(26,29)(28,32)(30,31);(3,7)(5,8)(11,16)(13,14)(17,18,21,20)(19,22,23,24)(25,27,28,32)(26,30,31,29);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,31)(29,32);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,18)(19,24,23,22)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2148187);