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 | #45104 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,7)(3,4,8,5)(10,13)(14,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,30)(28,31)(29,32);(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32);(2,7)(3,8)(18,20)(19,24)(25,28)(27,32);(1,3)(2,4)(5,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29);(1,4,6,5)(2,8,7,3)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32);(1,5,6,4)(2,3,7,8)(17,22,23,31)(18,25,20,28)(19,27,24,32)(21,29,26,30)Polycyclic presentation:
2^9|a^2=hi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7603404);