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 | |
4964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,28)(21,23)(22,24)(25,30)(27,31)(29,32);(1,2,5,4)(3,7,8,6)(10,13)(14,15)(17,19,22,21)(18,23,24,26)(20,27,30,29)(25,31,28,32);(2,4)(6,7)(9,10)(11,14)(12,13)(15,16)(18,24)(23,26)(25,28)(31,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,28)(26,32);(17,21,22,19)(18,26,24,23)(20,29,30,27)(25,32,28,31);(1,4,5,2)(3,7,8,6)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|b^2=gi,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103109);