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 | |
4742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(10,13)(14,15)(17,18)(19,24)(20,26)(21,22)(23,25)(27,31)(28,32)(29,30);(1,2,5,6)(3,7,4,8)(17,19)(18,24)(20,27)(21,30)(22,29)(23,28)(25,32)(26,31);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,13)(14,16)(18,25)(19,28)(20,30)(22,31);(9,11,12,16)(10,14,13,15)(17,20,23,30)(18,26,25,29)(19,27,28,21)(22,24,31,32);(1,3,5,4)(2,7,6,8)(17,21,23,27)(18,22,25,31)(19,20,28,30)(24,26,32,29);(1,4,5,3)(2,8,6,7)(17,22)(18,27)(19,29)(20,24)(21,25)(23,31)(26,28)(30,32)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,e^2=gi,f^2=g,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, 7590061);