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 | |
4899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(4,8)(6,7)(9,10,13,16)(11,15,12,14)(17,18)(19,23)(20,28)(21,26)(22,24)(25,30)(27,29)(31,32);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,14,16,15)(17,19,22,26)(18,23,24,21)(20,27,30,32)(25,31,28,29);(2,3)(6,7)(9,10,13,16)(11,14,12,15)(18,24)(19,26)(25,28)(27,32);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,25)(22,24)(23,26)(27,29)(28,30)(31,32);(17,20,22,30)(18,25,24,28)(19,27,26,32)(21,29,23,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,15,16,14)(17,21)(18,19)(20,29)(22,23)(24,26)(25,27)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014706);