Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2337 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)(10,14)(15,16)(17,18,22,25)(19,21,27,23)(20,28,31,24)(26,30,29,32);(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,27)(18,23,25,21)(20,29,31,26)(24,32,28,30);(5,6)(7,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,32)(29,31);(1,2,3,4)(9,10)(11,15)(12,16)(13,14)(17,20)(18,24)(19,26)(21,32)(22,31)(23,30)(25,28)(27,29);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,19,25,27)(20,30,31,32)(24,29,28,26)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2828325);