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 | |
2130 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)(11,15)(12,14)(13,16)(17,18,21,22)(19,25,26,23)(20,24,28,30)(27,32,29,31);(6,8)(9,11,14,13)(10,15,12,16)(18,22)(19,26)(20,28)(31,32);(5,6,7,8)(9,11,14,13)(10,16,12,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31);(1,2)(3,4)(9,12,14,10)(11,15,13,16)(17,19)(18,23)(20,29)(21,26)(22,25)(24,32)(27,28)(30,31);(9,13,14,11)(10,15,12,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=gh,d^2=fg,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, 2978802);