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 | |
4743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,12)(14,16)(17,18)(19,25)(20,22)(21,24)(23,26)(27,32)(28,30)(29,31);(1,2,4,5)(3,6,7,8)(11,15)(14,16)(18,20)(19,26)(24,28)(27,29);(2,5)(6,8)(9,10,13,12)(11,14,15,16)(18,20)(23,25)(24,28)(31,32);(3,7)(6,8)(9,11,13,15)(10,14,12,16)(17,19,22,26)(18,23,20,25)(21,29,30,27)(24,32,28,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,20,22,18)(19,25,26,23)(21,28,30,24)(27,32,29,31);(1,3,4,7)(2,6,5,8)(17,21,22,30)(18,24,20,28)(19,27,26,29)(23,31,25,32)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8543490);