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) = 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 | |
2166 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)(11,13)(12,14)(15,16)(17,18,23,26)(19,24,21,20)(22,30,31,25)(27,28,29,32);(6,8)(9,10,12,14)(11,15,16,13)(17,19,23,21)(18,20,26,24)(22,29,31,27)(25,32,30,28);(5,6)(7,8)(9,10,12,14)(11,13,16,15)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32);(1,2,3,4)(9,10)(11,15)(12,14)(13,16)(17,21,23,19)(18,24,26,20)(22,27,31,29)(25,28,30,32);(9,11)(10,13)(12,16)(14,15)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)Polycyclic presentation:
2^9|a^2=fg,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, 3042668);