Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2342 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)(9,10,12,14)(11,15,16,13)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,29)(31,32);(6,8)(9,10,12,14)(11,13,16,15)(18,23)(19,26)(25,28)(27,32);(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,29)(28,31)(30,32);(1,2,3,4)(9,10)(11,13)(12,14)(15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,31,24,29);(9,11)(10,13)(12,16)(14,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,28,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=gh,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3373364);