Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,21,23)(19,20,24,22)(27,31)(30,32);(1,2,4,6)(3,5,7,8)(9,11)(10,15)(12,16)(13,14)(17,18,21,23)(19,20,24,22)(26,29)(30,32);(1,3,4,7)(2,5,6,8)(9,10,14,12)(11,16,13,15)(17,19,21,24)(18,22,23,20)(25,26,28,29)(27,30,31,32);(1,3)(2,5)(4,7)(6,8)(9,12,14,10)(11,16,13,15)(17,19)(18,20)(21,24)(22,23);(1,2)(3,5)(4,6)(7,8)(9,13,14,11)(10,15,12,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fhi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5626401);