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 | |
4721 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)(3,5)(4,8)(6,7)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,21)(22,24)(25,28)(26,27)(29,30)(31,32);(1,3)(2,5)(4,6)(7,8)(10,11)(14,15)(17,19,22,28)(18,23,24,25)(20,29,27,31)(21,30,26,32);(1,2)(3,8)(4,5)(6,7)(9,11,13,10)(12,15,16,14)(18,24)(19,28)(21,26)(29,31);(2,7)(3,4)(9,12,13,16)(10,14,11,15)(17,19)(18,25)(20,31)(21,30)(22,28)(23,24)(26,32)(27,29);(1,4,6,3)(2,5,7,8)(17,20,22,27)(18,26,24,21)(19,29,28,31)(23,32,25,30);(1,5)(2,3)(4,7)(6,8)(17,21,22,26)(18,27,24,20)(19,30,28,32)(23,31,25,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=gi,f^2=h,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, 8841938);