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 | |
4797 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,22)(23,25)(24,28)(26,30)(27,31)(29,32);(1,2,6,8)(3,5,7,4)(10,13)(14,15)(17,19,23,28)(18,24,25,21)(20,26,29,31)(22,27,32,30);(1,3,6,7)(2,4,8,5)(9,10)(11,14)(12,13)(15,16)(20,29)(22,32)(26,31)(27,30);(1,4,6,5)(2,3,8,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,22)(19,26)(21,30)(23,29)(24,27)(25,32)(28,31);(1,2)(3,4)(5,7)(6,8)(17,21,23,24)(18,19,25,28)(20,27,29,30)(22,31,32,26);(1,5,6,4)(2,7,8,3)(17,22)(18,20)(19,27)(21,31)(23,32)(24,26)(25,29)(28,30)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839406);