Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,24)(22,23)(26,30)(28,32);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,16,15)(17,19,21,23)(18,22,24,20)(25,26)(27,28)(29,30)(31,32);(3,8)(5,6)(9,11,13,12)(10,15,16,14)(17,20,21,22)(18,23,24,19)(25,27,29,31)(26,28,30,32);(1,2)(3,5)(4,7)(6,8)(9,12,13,11)(10,14,16,15)(17,20)(18,19)(21,22)(23,24)(25,28,29,32)(26,27,30,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4078695);