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 | |
5266 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)(3,7)(4,6)(5,8)(10,13)(14,16)(18,23)(19,25)(24,28)(27,31);(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,29)(31,32);(1,2,4,6)(3,5,8,7)(18,23)(21,26)(24,28)(29,32);(1,2)(3,5)(4,6)(7,8)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,22,30)(18,24,23,28)(19,27,25,31)(21,29,26,32);(17,21)(18,19)(20,29)(22,26)(23,25)(24,27)(28,31)(30,32)Polycyclic presentation:
2^9|c^2=h,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, 7713991);