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 | |
4981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,24)(19,27,28,23)(20,31,30,25)(21,26,29,32);(1,2,5,6)(3,7,4,8)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,32,31,26);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(18,24)(19,28)(20,30)(26,32);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,32)(30,31);(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,29)(21,28)(22,30)(23,32)(24,31)(26,27);(1,4,5,3)(2,8,6,7)(17,21,22,29)(18,26,24,32)(19,30,28,20)(23,31,27,25)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7881074);