Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,8)(9,10,14,16)(11,15,12,13)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31);(9,11,14,12)(10,13,16,15)(19,26)(22,25)(27,32)(30,31);(5,6,7,8)(9,12,14,11)(10,15,16,13)(17,19,21,26)(18,22,24,25)(20,27,29,32)(23,30,28,31);(1,2,3,4)(5,6)(7,8)(9,13,14,15)(10,12,16,11)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32);(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=g,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837872);