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 | |
4945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,7)(9,10,13,11)(12,15,16,14)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,30)(31,32);(1,2,5,4)(3,6,8,7)(10,11)(14,15)(17,19,23,26)(18,22,27,28)(20,29,25,32)(21,30,24,31);(1,2,5,4)(3,7,8,6)(9,11,13,10)(12,15,16,14)(17,20,23,25)(18,24,27,21)(19,29,26,32)(22,31,28,30);(1,3,5,8)(2,6,4,7)(9,12)(10,14)(11,15)(13,16)(17,20)(18,24)(19,29)(21,27)(22,31)(23,25)(26,32)(28,30);(1,3)(2,7)(4,6)(5,8)(17,21,23,24)(18,25,27,20)(19,30,26,31)(22,32,28,29);(1,4,5,2)(3,7,8,6)(17,22)(18,26)(19,27)(20,31)(21,29)(23,28)(24,32)(25,30)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113726);