Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2510 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)(9,10,12,13)(11,15,16,14)(17,18)(19,26)(20,21)(22,24)(23,25)(27,30)(28,29)(31,32);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,32,30);(5,6)(7,8)(10,13)(14,15)(17,20,23,19)(18,21,25,26)(22,27,31,28)(24,30,32,29);(1,2,3,4)(9,11,12,16)(10,14,13,15)(17,21)(18,19)(20,25)(22,30)(23,26)(24,28)(27,32)(29,31);(9,10)(11,15)(12,13)(14,16)(17,22,23,31)(18,24,25,32)(19,27,20,28)(21,29,26,30)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fg,d^2=fgi,e^2=fg,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, 3372567);