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 | |
2570 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)(10,11)(12,16)(17,18,23,26)(19,20,21,24)(22,30,32,25)(27,31,29,28);(6,8)(10,11)(14,15)(17,19,23,21)(18,24,26,20)(22,27,32,29)(25,31,30,28);(5,6,7,8)(9,10,13,11)(12,14,16,15)(17,20,23,24)(18,19,26,21)(22,31,32,28)(25,29,30,27);(1,2,3,4)(9,11,13,10)(12,15,16,14)(17,21,23,19)(18,24,26,20)(22,27,32,29)(25,28,30,31);(9,12)(10,14)(11,15)(13,16)(17,22,23,32)(18,25,26,30)(19,27,21,29)(20,28,24,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=fi,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, 3373124);