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 | |
4882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,7,5,4)(11,16)(12,14)(17,18,22,26)(19,23,27,31)(20,29,30,24)(21,28,32,25);(1,3,6,5)(2,4,8,7)(9,10)(11,12)(13,15)(14,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,31)(28,30);(1,4)(2,5)(3,8)(6,7)(9,11)(10,14)(12,15)(13,16)(19,27)(20,30)(23,31)(24,29);(3,5)(4,7)(9,11,13,16)(10,14,15,12)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,5,6,3)(2,7,8,4)(9,11)(10,12)(13,16)(14,15)(17,21)(18,25)(19,24)(20,31)(22,32)(23,30)(26,28)(27,29);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=hi,b^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841052);