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 | |
5224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(19,25)(23,26)(27,31)(30,32);(1,2,4,6)(3,5,7,8)(10,14)(11,15)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32);(1,3)(2,5)(4,7)(6,8)(9,11,13,15)(10,12,14,16)(18,22)(19,25)(20,28)(30,32);(9,12)(10,15)(11,14)(13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,2)(3,5)(4,6)(7,8)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3)(2,5)(4,7)(6,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8957601);