Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(10,14)(11,15)(17,18)(19,20)(21,23)(22,24)(25,26,28,30)(27,31,32,29);(2,3)(6,7)(9,10,13,14)(11,16,15,12)(17,19,22,23)(18,21,24,20)(27,32)(29,31);(4,8)(6,7)(10,14)(12,16)(17,20,22,21)(18,19,24,23)(25,27)(26,29)(28,32)(30,31);(1,3,5,2)(4,6,8,7)(9,11,13,15)(10,12,14,16)(17,21,22,20)(18,23,24,19)(25,26)(27,29)(28,30)(31,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,21)(20,23)(22,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=hi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3972838);