Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2837 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)(19,24,20,21)(22,25,31,32)(27,30,28,29);(6,8)(10,11)(14,15)(17,19,23,20)(18,21,26,24)(22,28,31,27)(25,30,32,29);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,30,32,29);(1,2,3,4)(9,11,13,10)(12,15,16,14)(17,21,23,24)(18,20,26,19)(22,30,31,29)(25,27,32,28);(9,12,13,16)(10,14,11,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3042636);