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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30);(9,11,13,16)(10,14,12,15)(17,19,23,26)(18,21,20,25)(22,30,31,27)(24,32,28,29);(2,6)(3,7)(18,20)(19,26)(22,31)(29,32);(1,3)(2,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20,23,18)(19,21,26,25)(22,28,31,24)(27,29,30,32);(1,4)(2,3)(5,8)(6,7)(17,21)(18,19)(20,26)(22,29)(23,25)(24,27)(28,30)(31,32);(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,32)(26,30)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713840);