Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2286 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)(9,10,14,12)(11,16,15,13)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,31,30,32);(6,8)(9,10,14,12)(11,16,15,13)(18,22)(20,29)(23,25)(26,30);(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,27)(18,23,22,25)(20,30,29,26)(24,32,28,31);(1,2)(3,4)(9,12,14,10)(11,16,15,13)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,32)(30,31);(9,13,14,16)(10,15,12,11)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3941564);