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 | |
4742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(4,8)(6,7)(9,10,13,11)(12,15,16,14)(18,23)(24,30)(25,26)(31,32);(1,2,5,3)(4,6,8,7)(10,11)(14,15)(18,23)(19,27)(25,26)(28,29);(2,3)(6,7)(9,11,13,10)(12,15,16,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,31)(29,32);(9,12)(10,14)(11,15)(13,16)(17,19,22,27)(18,24,23,30)(20,28,21,29)(25,31,26,32);(1,3,5,2)(4,7,8,6)(17,20,22,21)(18,25,23,26)(19,28,27,29)(24,31,30,32);(1,4)(2,6)(3,7)(5,8)(17,21,22,20)(18,26,23,25)(19,29,27,28)(24,32,30,31)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002169);