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 | |
5692 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,30)(21,25)(22,24)(23,27)(28,29)(31,32);(1,3,5,4)(2,6,7,8)(17,19,22,27)(18,23,24,26)(20,31,25,28)(21,32,30,29);(2,7)(6,8)(9,10,12,14)(11,13,16,15)(18,24)(19,27)(21,30)(28,31);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,31)(30,32);(1,4,5,3)(2,8,7,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,25)(18,21,24,30)(19,28,27,31)(23,29,26,32);(17,21)(18,25)(19,29)(20,24)(22,30)(23,31)(26,28)(27,32)Polycyclic presentation:
2^9|b^2=g,c^2=gh,e^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, 7713997);