Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(10,14)(15,16)(17,18,21,24)(19,22,23,20);(2,6)(4,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,23)(18,22,24,20)(26,29)(30,32);(1,3)(2,4)(5,8)(6,7)(9,11,13,12)(10,15,14,16)(17,20,21,22)(18,23,24,19)(25,26)(27,30)(28,29)(31,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(25,27,28,31)(26,30,29,32);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,15,14,16)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1345268);