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 | |
5186 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)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18)(19,22)(20,28)(21,23)(24,26)(25,30)(27,32)(29,31);(9,10,12,13)(11,14,15,16)(17,19,21,26)(18,22,23,24)(20,29,30,27)(25,32,28,31);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(18,23)(19,26)(25,28)(27,29);(1,3,5,7)(2,4,6,8)(17,19)(18,24)(20,27)(21,26)(22,23)(25,32)(28,31)(29,30);(1,4)(2,7)(3,6)(5,8)(17,20,21,30)(18,25,23,28)(19,27,26,29)(22,31,24,32);(17,18)(19,24)(20,25)(21,23)(22,26)(27,32)(28,30)(29,31)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=hi,e^2=i,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, 7717788);