Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #44927 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,6,8,4)(10,14)(15,16)(17,18,23,20)(19,25,26,21)(22,28,30,24)(27,29,32,31);(9,10)(11,15)(12,16)(13,14)(17,19)(18,21)(20,25)(22,27)(23,26)(24,29)(28,31)(30,32);(1,3,5,8)(2,4,7,6)(9,11,13,12)(10,15,14,16)(18,20)(19,26)(22,30)(29,31);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20,23,18)(19,25,26,21)(22,28,30,24)(27,31,32,29);(17,21,23,25)(18,19,20,26)(22,29,30,31)(24,27,28,32);(1,4,5,6)(2,3,7,8)(9,12,13,11)(10,16,14,15)(17,22)(18,24)(19,27)(20,28)(21,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7590420);