Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(4,8)(9,10,14,13)(11,16,15,12)(19,23)(22,24);(2,3)(6,7)(9,11)(10,12)(13,16)(14,15)(18,21)(22,24)(25,26,28,30)(27,29,31,32);(1,2,5,3)(4,6,8,7)(9,12,14,16)(10,15,13,11)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,15)(14,16)(17,19,20,23)(18,22,21,24)(25,27,28,31)(26,29,30,32);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,16,15,12)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2675354);