Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,8)(4,6)(10,12)(11,15)(18,19)(20,24)(25,26)(27,31)(28,29)(30,32);(1,2)(3,4)(5,7)(6,8)(9,10,14,12)(11,16,15,13)(17,18,21,19)(20,23,24,22)(25,27,30,29)(26,31,32,28);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(17,19,21,18)(20,23,24,22)(25,28)(26,27)(29,32)(30,31);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,16,15,13)(25,29,30,27)(26,31,32,28);(1,4)(2,3)(5,6)(7,8)(9,13,14,16)(10,15,12,11)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|b^2=fhi,c^2=fh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4861956);