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 | |
5000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,15,16)(11,13,14,12)(17,18,23,22)(19,20,24,27)(21,29,30,25)(26,32,31,28);(1,2,5,6)(3,7,8,4)(9,11,15,14)(10,12,16,13)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31);(2,6)(4,7)(9,12,15,13)(10,11,16,14)(18,22)(19,24)(25,29)(26,31);(1,3)(2,4)(5,8)(6,7)(9,13,15,12)(10,11,16,14)(17,20,23,27)(18,24,22,19)(21,28,30,32)(25,31,29,26);(17,21,23,30)(18,25,22,29)(19,26,24,31)(20,28,27,32);(1,4)(2,3)(5,7)(6,8)(9,14,15,11)(10,12,16,13)(17,22,23,18)(19,27,24,20)(21,29,30,25)(26,32,31,28)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=gh,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, 8090107);