Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(9,10)(11,14)(12,16)(13,15)(17,18,22,23)(19,24,26,21)(20,28,31,25)(27,32,29,30);(6,8)(9,11,13,16)(10,14,15,12)(18,23)(21,24)(25,28)(30,32);(5,6,7,8)(9,12)(10,11)(13,14)(15,16)(17,19,22,26)(18,24,23,21)(20,29,31,27)(25,32,28,30);(1,2,3,4)(9,11)(10,12)(13,16)(14,15)(17,20)(18,25)(19,27)(21,32)(22,31)(23,28)(24,30)(26,29);(9,12)(10,11)(13,14)(15,16)(17,21,22,24)(18,26,23,19)(20,30,31,32)(25,27,28,29)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=fgi,e^2=f,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, 3043446);