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 | |
5088 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)(3,4)(5,7)(6,8)(9,10,13,15)(11,12,16,14)(17,18,22,26)(19,27,29,23)(20,30,28,24)(21,25,31,32);(3,8)(4,6)(9,11,13,16)(10,14,15,12)(17,19,22,29)(18,23,26,27)(20,21,28,31)(24,25,30,32);(1,3,5,8)(2,6,7,4)(9,11)(10,12)(13,16)(14,15);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,21)(22,28)(23,25)(26,30)(27,32)(29,31);(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,8)(5,7)(17,21)(18,25)(19,28)(20,29)(22,31)(23,30)(24,27)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,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, 8337032);