Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,5,8,7)(9,10,14,12)(11,16,13,15)(17,18,21,19)(20,22,24,23)(27,31)(29,32);(1,3,6,8)(2,7,4,5)(9,10,14,12)(11,15,13,16)(18,19)(20,24)(25,26,28,30)(27,29,31,32);(2,4)(5,7)(9,11,14,13)(10,15,12,16)(17,19,21,18)(20,22,24,23);(1,4,6,2)(3,5,8,7)(9,12,14,10)(11,15,13,16)(25,27)(26,29)(28,31)(30,32);(1,5)(2,3)(4,8)(6,7)(9,13,14,11)(10,15,12,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=fg,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393459);