Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| 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 | |
2011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,5,4,8)(10,11)(14,15)(17,18,22,23)(19,20,21,24)(25,26,30,29)(27,32,31,28);(2,7)(5,8)(9,10,13,11)(12,15,16,14)(18,23)(19,21);(1,3,6,4)(2,5,7,8)(9,10,13,11)(12,14,16,15)(17,19,22,21)(18,24,23,20)(25,27)(26,28)(29,32)(30,31);(1,4,6,3)(2,5,7,8)(9,11,13,10)(12,14,16,15)(17,20)(18,19)(21,23)(22,24)(25,28,30,32)(26,31,29,27);(1,5)(2,4)(3,7)(6,8)(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,24,23,20)(25,29,30,26)(27,32,31,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=fghi,c^2=f,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 736713);