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 | |
5033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,7)(3,5,8,6)(9,10,13,15)(11,12,16,14)(17,18,23,26)(19,27,30,24)(20,31,28,22)(21,25,32,29);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,15,12)(17,19,23,30)(18,24,26,27)(20,32,28,21)(22,29,31,25);(1,3,4,8)(2,6,7,5)(9,11)(10,12)(13,16)(14,15)(20,28)(21,32)(22,31)(25,29);(3,8)(5,6)(9,12)(10,11)(13,14)(15,16)(17,20)(18,22)(19,21)(23,28)(24,25)(26,31)(27,29)(30,32);(1,3)(2,5)(4,8)(6,7)(17,21)(18,25)(19,28)(20,30)(22,27)(23,32)(24,31)(26,29);(1,2)(3,5)(4,7)(6,8)(17,22)(18,20)(19,29)(21,27)(23,31)(24,32)(25,30)(26,28)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965402);