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 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18,24,27)(19,28,25,20)(21,31,32,23)(22,26,30,29);(1,2)(3,6)(4,7)(5,8)(10,13)(11,15)(17,19,24,25)(18,20,27,28)(21,30,32,22)(23,29,31,26);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20,24,28)(18,25,27,19)(21,26,32,29)(22,23,30,31);(9,11)(10,14)(12,15)(13,16)(17,21)(18,23)(19,22)(20,26)(24,32)(25,30)(27,31)(28,29);(1,3)(2,6)(4,8)(5,7)(17,22)(18,26)(19,21)(20,23)(24,30)(25,32)(27,29)(28,31);(1,2)(3,5)(4,7)(6,8)(17,23)(18,21)(19,29)(20,30)(22,28)(24,31)(25,26)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003058);