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 | |
4841 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)(4,5)(6,8)(9,10)(11,13)(12,15)(14,16)(18,24)(19,27)(20,29)(26,32);(1,3)(2,7)(4,8)(5,6)(10,11)(14,15)(17,18,23,24)(19,25,27,31)(20,30,29,22)(21,26,28,32);(1,4)(2,3)(5,8)(6,7)(9,10,13,11)(12,14,16,15)(17,19)(18,25)(20,28)(21,29)(22,26)(23,27)(24,31)(30,32);(1,5,6,3)(2,4,8,7)(9,11,13,10)(12,15,16,14)(17,20,23,29)(18,22,24,30)(19,28,27,21)(25,26,31,32);(9,12,13,16)(10,14,11,15)(17,21,23,28)(18,26,24,32)(19,29,27,20)(22,31,30,25);(1,4)(2,5)(3,8)(6,7)(17,22)(18,20)(19,26)(21,31)(23,30)(24,29)(25,28)(27,32)Polycyclic presentation:
2^9|b^2=hi,c^2=ghi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343760);