Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,8)(10,13)(14,16)(17,18)(19,24)(20,23)(21,27)(22,28)(25,30)(26,32)(29,31);(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,24,20,27)(22,26,30,29)(25,31,28,32);(5,6)(7,8)(17,20,23,18)(19,24,21,27)(22,28,30,25)(26,31,29,32);(1,2,3,4)(9,10)(11,14)(12,13)(15,16)(17,21,23,19)(18,24,20,27)(22,29,30,26)(25,31,28,32);(9,11,12,15)(10,14,13,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215623);